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# FORCE

 In physics, a force is any influence that causes a free body to undergo an acceleration. Force can also be described by intuitive concepts such as a push or pull that can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate, or which can cause a flexible object to deform. A force has both magnitude and direction, making it a vector quantity. Newton's second law, F=ma, can be formulated to state that an object with a constant mass will accelerate in proportion to the net force acting upon and in inverse proportion to its mass, an approximation which breaks down near the speed of light. Newton's original formulation is exact, and does not break down: this version states that the net force acting upon an object is equal to the rate at which its momentum changes.[1]Related concepts to accelerating forces include thrust, increasing the velocity of the object, drag, decreasing the velocity of any object, and torque, causingchanges in rotational speed about an axis. Forces which do not act uniformly on all parts of a body will also cause mechanical stresses,[2] a technical term for influences which cause deformation of matter. While mechanical stress can remain embedded in a solid object, gradually deforming it, mechanical stress in a fluid determines changes in its pressure and volume.[3][4]Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle andArchimedes retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of friction, and a consequently inadequate view of the nature of natural motion[5] A fundamental error was the belief that a force is requied to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Sir Isaac Newton; with his mathematical insight, he formulated laws of motion that remained unchanged for nearly three hundred years.[4] By the early 20th century, Einstein developed atheory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light, and also provided insight into the forces produced by gravitation and inertia.With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised aStandard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchange particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong,electromagnetic, weak, and gravitational.[3] High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.

# FORCE

In physics, force is action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force). In an extended body, force may also cause rotation, deformation, or an increase in pressure for the body. Rotational effects are determined by the torques, while deformation and pressure are determined by the stresses that the forces create.

Net force is mathematically equal to the rate of change of the momentum of the body on which it acts. Since momentum is a vector quantity (has both a magnitude and direction), force also is a vector quantity.

The concept of force has formed part of statics and dynamics since ancient times. Ancient contributions to statics culminated in the work of Archimedes in the 3rd century BC, which still forms part of modern physics. In contrast, Aristotle's dynamics incorporated intuitive misunderstandings of the role of force which were eventually corrected in the 17th century, culminating in the work of Isaac Newton. Following the development of quantum mechanics it is now understood that particles influence each another through fundamental interactions, making force a useful concept only on the macroscopic level. Only four fundamental interactions are known: strong, electromagnetic, weak (unified into one electroweak interaction in 1970s), and gravitational (in order of decreasing strength).

## History

Aristotle and his followers believed that it was the natural state of objects on Earth to be motionless and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g. for heavy bodies to fall), which lead to "natural motion", and unnatural or forced motion, which required continued application of a force. But this theory, although based on the everyday experience of how objects move (e.g. a horse and cart), had severe trouble accounting for projectiles, such as the flight of arrows. Several theories were discussed over the centuries, and the late medieval idea that objects in forced motion carried an innate force of impetus was influential on the work of Galileo. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the 17th century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force - usually friction.

Isaac Newton is recognised as having argued explicitly for the first time that, in general, a constant force causes a constant rate of change (time derivative) of momentum. In essense, he gave the first (and the only) mathematical definition of the quantity force itself - as being the time-derivative of momentum: F = dp/dt.

In 1784 Charles Coulomb discovered the inverse square law of interaction between electric charges using a torsion balance, which was the second fundamental force. The weak and strong forces were discovered in the 20th century.

With the development of quantum field theory and general relativity it was realized that “force” is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in QED). Thus currently known fundamental forces are more accurately called “fundamental interactions”.

## Types of force

Although there are apparently many types of forces in the Universe, they are all based on four fundamental forces. The strong and weak forces only act at very short distances and are responsible for holding certain nucleons and compound nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. The Pauli exclusion principle is responsible for the tendency of atoms not to overlap each other, and is thus responsible for the "stiffness" or "rigidness" of matter, but this also depends on the electromagnetic force which binds the constituents of every atom.

All other forces are based on these four. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli exclusion principle, which does not allow atoms to pass through each other. The forces in springs modeled by Hooke's law are also the result of electromagnetic forces and the exclusion principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.

There is currently some debate to whether there are five forces not four, due to the discovery of dark energy, which could be just an energy of vacuum fluctuations, or it could be a new type of energy resulting in a repulsive force.

The modern quantum mechanical view of the first three fundamental forces (all except gravity) is that particles of matter (fermions) do not directly interact with each other but rather by exchange of virtual particles (bosons). This exchange results in what we call electromagnetic interaction (Coulomb force is one example of electromagnetic interaction).

In general relativity, gravitation is not viewed as a force. Rather, objects moving freely in gravitational fields simply undergo inertial motion along a straight line in the curved space-time - defined as the shortest space-time path between two space-time points. This straight line in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola shape as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the body is what we label as "gravitational force".

## Examples

• A heavy object is in free fall. Its momentum changes as dp/dt = mdv/dt = ma =mg (if the mass m is constant), thus we call the quantity mg a "gravitational force" acting on the object. This is the definition of weight (w=mg) of an object.
• A heavy object on a table is pulled (attracted) downward toward the floor by the force of gravity (i.e., its weight). At the same time, the table resists the downward force with equal upward force (called the normal force), resulting in zero net force, and no acceleration. (If the object is a person, he actually feels the normal force acting on him from below.)
• A heavy object on a table is gently pushed in a sideways direction by a finger. However, it doesn't move because the force of the finger on the object is now opposed by a new force of static friction, generated between the object and the table surface. This newly generated force exactly balances the force exerted on the object by the finger, and again no acceleration occurs. The static friction increases or decreases automatically. If the force of the finger is increased (up to a point), the opposing sideways force of static friction increases exactly to the point of perfect opposition.
• A heavy object on a table is pushed by a finger hard enough that static friction cannot generate sufficient force to match the force exerted by the finger, and the object starts sliding across the surface. If the finger is moved with a constant velocity, it needs to apply a force that exactly cancels the force of kinetic friction from the surface of the table and then the object moves with the same constant velocity. Here it seems to the naive observer that application of a force produces a velocity (rather than an acceleration). However, the velocity is constant only because the force of the finger and the kinetic friction cancel each other. Without friction, the object would continually accelerate in response to a constant force.
• A heavy object reaches the edge of the table and falls. Now the object, subjected to the constant force of its weight, but freed of the normal force and friction forces from the table, gains in velocity in direct proportion to the time of fall, and thus (before it reaches velocities where air resistance forces becomes significant compared to gravity forces) its rate of gain in momentum and velocity is constant. These facts were first discovered by Galileo.
• A heavy object is suspended on a spring scale. Because object is not moving (so time derivative of its momentum is zero) then along with acceleration of free fall g it has to experience equal and oppositely directed acceleration a=-g due to the action of the spring. This acceleration multiplied by the mass of the object is what we label as "spring reaction force" which is obviousely equal and opposite to object's weight mg. Knowing the mass (say, 1 kg) and the acceleration of free fall (say, 9.80 m/s^2) we can calibrate spring scale by a mark "9.8 N". Attaching various masses (2 kg, 3 kg ...) we can calibrate spring scale and then use this calibrated scale to measure many other forces (friction, reaction forces, elecric force, magnetic force, etc).

## Quantitative definition

We have an intuitive grasp of the notion of force, since forces can be directly perceived as a push or pull. As with other physical concepts (e.g. temperature), the intuitive notion is quantified using operational definitions that are consistent with direct perception, but more precise. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces cancelled each other out. Such experiments prove the crucial properties that forces are additive vector quantities: they have magnitude and direction. So, when two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. As with all vector addition this results in a parallelogram rule: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram.

As well as being added, forces can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.

The simplest case of static equilibrium is when two forces are equal in magnitude but opposite in direction. This remains the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. Using such tools, several quantitative force laws were discovered: that the force of gravity is proportional to volume for objects made of a given material (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs: all these were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.

Force is sometimes defined using Newton's second law, as the product of mass m times acceleration $\vec{a}$:

$\vec{F} = m \vec{a}$

(or, more generally, as the rate of change of momentum). This approach is disparaged by the large majority of textbooks. By making this a definition of force, all empirical content is removed from the law. In fact, the $\vec{F}$ in this equation represents the net (vector sum) force; in static equilibrium this is zero by definition, but (balanced) forces are present nevertheless. Instead, Newton's law is meaningful because it asserts the proportionality of two quantities which can be defined without reference to it. Thus, the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity (therefore zero acceleration) is objectively wrong and not just a consequence of a poor choice definition. With rather more justification, Newton's second law can be taken as a quantitative definition of mass; certainly, by writing the law as an equality, the relative units of force and mass are fixed.

Given the empirical success of Newton's law, it is sometimes used to measure the strength of forces (for instance, using astronomical orbits to determine gravitational forces). Nevertheless, the force and the (mass times acceleration) used to measure it remain distinct concepts.

The definition of force is sometimes regarded as problematic, since it must either ultimately be referred to our intuitive understanding of our direct perceptions, or be defined implicitly through a set of self-consistent mathematical formulae. Notable physicists, philosophers and mathematicians who have sought a more explicit definition include Ernst Mach, Clifford Truesdell and Walter Noll.[2]

## Force in special relativity

In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate a body). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires a greater force to accelerate it the same amount than it did at a lower velocity. The definition $\vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t$ remains valid. But in order to be conserved, momentum must be redefined as:

$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$

where

v is the velocity and
c is the speed of light.

The relativistic expression relating force and acceleration for a particle with non-zero rest mass $m\,$ moving in the $x\,$ direction is:

$F_x = \gamma^3 m a_x \,$
$F_y = \gamma m a_y \,$
$F_z = \gamma m a_z \,$

where the Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that γ is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed.

One can however restore the form of

$F^\mu = mA^\mu \,$

for use in relativity through the use of four-vectors. This relation is correct in relativity when Fμ is the four-force, m is the invariant mass, and Aμ is the four-acceleration.

## Force and potential

Instead of a force, the mathematically equivalent concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. Restating mathematically the definition of energy (via definition of work), a potential scalar field $U(\vec{r})$ is defined as that field whose gradient is equal and opposite to the force produced at every point:

$\vec{F}=-\vec{\nabla} U$

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential.

### Conservative forces

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces, therefore, have models which are dependent on a position often given as a radial vector $\vec{r}$ eminating from spherically symmetric potentials. Examples of this follow:

For gravity:

$\vec{F} = - \frac{G m_1 m_2 \vec{r}}{r^3}$

where G is the gravitational constant, mn is the mass of object n.

For electrostatic forces:

$\vec{F} = - \frac{q_{1} q_{2} \vec{r}}{4 \pi \epsilon_{0} r^3}$

where ε0 is electric permittivity of free space, qn is the electric charge of object n.

For spring forces:

$\vec{F} = - k \vec{r}$

where k is the spring constant.

### Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations which yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model which is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.

The connection between macroscopic non-conservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system and are often associated with the transfer of heat. According to the Second Law of Thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.

## Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s−2. The earlier CGS unit is the dyne. The relationship F=m·a can be used with either of these. In Imperial engineering units, if F is measured in "pounds force" or "lbf", and a in feet per second squared, then m must be measured in slugs. Similarly, if mass is measured in pounds mass, and a in feet per second squared, the force must be measured in poundals. The units of slugs and poundals are specifically designed to avoid a constant of proportionality in this equation.

A more general form F=k·m·a is needed if consistent units are not used. Here, the constant k is a conversion factor dependent upon the units being used.

When the standard g (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf.

The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard g which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity.

By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the hyl, TME (from a German acronym), and mug (from metric slug).

Another unit of force called the poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl. The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern SI system, but is still used in applications such as:

• Thrust of jet and rocket engines
• Spoke tension of bicycles
• Draw weight of bows
• Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
• Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
• Pressure gauges in "kg/cm²" or "kgf/cm²"

In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force.

The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to distinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.

### Conversions

Below are several conversion factors between various measurements of force:

• 1 dyne = 10-5 newtons
• 1 kgf (kilopond kp) = 9.80665 newtons
• 1 metric slug = 9.80665 kg
• 1 lbf = 32.174 poundals
• 1 slug = 32.174 lb
• 1 kgf = 2.2046 lbf

# FORCE

Forces are often described as pushes or pulls. They can be due to phenomena such as gravity, magnetism, or anything else that causes a mass to accelerate.
Classical mechanics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law
History of ...
Fundamental concepts
Space · Time · Mass · Force
Energy · Momentum
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In physics, a force is whatever can cause an object with mass to accelerate.[1] Force has both magnitude and direction, making it a vector quantity. According to Newton's second law, an object with constant mass will accelerate in proportion to the net force acting upon it and in inverse proportion to its mass. An equivalent formulation is that the net force on an object is equal to the rate of change of momentum it experiences.[2] Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure. The tendency of a force to cause angular acceleration about an axis is termed torque. Deformation and pressure are the result of stress forces within an object.[3][4]

Since antiquity, scientists have used the concept of force in the study of stationary and moving objects. These studies culminated with the descriptions made by the third century BC philosopher Archimedes of how simple machines functioned. The rules Archimedes determined for how forces interact in simple machines are still a part of physics.[5] Earlier descriptions of forces by Aristotle incorporated fundamental misunderstandings which would not be corrected until the seventeenth century by Isaac Newton.[4] Newtonian descriptions of forces remained unchanged for nearly three hundred years.

Current understanding of quantum mechanics and the standard model of particle physics associates forces with the fundamental interactions accompanying the emission or absorption of gauge bosons. Only four fundamental interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational.[3] High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a unified electroweak interaction.[6] Einstein in his theory of general relativity explained that gravity is an attribute of the curvature of space-time.

Pre-Newtonian concepts

Aristotle famously described a force as anything which causes an object to undergo "unnatural motion"

Since antiquity, the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.[5]

Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the natural world held four elements that existed in "natural states". Aristotle believed that it was the natural state of objects with mass on Earth, such as the elements water and earth, to be motionless on the ground and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force.[7] This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. The place where forces were applied to projectiles was only at the start of the flight, and while the projectile sailed through the air, no discernible force acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path provided the needed force to continue the projectile moving. This explanation demands that air is needed for projectiles and that, for example, in a vacuum, no projectile would move after the initial push. Additional problems with the explanation include the fact that air resists the motion of the projectiles.[8]

These shortcomings would not be fully explained and corrected until the seventeenth century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the seventeenth century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.[9]

## Newtonian mechanics

Isaac Newton is the first person known to explicitly state the first, and the only, mathematical definition of force—as the time-derivative of momentum: $\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{dt}}$. In 1687, Newton went on to publish his Philosophiae Naturalis Principia Mathematica, which used concepts of inertia, force, and conservation to describe the motion of all objects.[10][4] In this work, Newton set out three laws of motion that to this day are the way forces are described in physics.[10]

Though Sir Isaac Newton's most famous equation is F=ma, he actually wrote down a different form for his second law of motion that used differential calculus.

### Newton's first law

Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted upon by an external net force or resultant force.[10] This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". That is, the first law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making rest physically indistinguishable from non-zero constant velocity, Newton's first law directly connects inertia with the concept of relative velocities. Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". In other words, to phrase matters more technically, the laws of physics are the same in every inertial frame of reference, that is, in all frames related by a Galilean transformation.

For example, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change from being at rest. A person can throw a ball straight up in the air and catch it as it falls down without worrying about applying a force in the direction the vehicle is moving. This is true even though another person who is observing the moving vehicle pass by also observes the ball follow a curving parabolic path in the same direction as the motion of the vehicle. It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and every thing inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Inertia therefore applies equally well to constant velocity motion as it does to rest.

The concept of inertia can be further generalized to explain the tendency of objects to continue in many different forms of constant motion, even those that are not strictly constant velocity. The rotational inertia of planet Earth is what fixes the constancy of the length of a day and the length of a year. Albert Einstein extended the principle of inertia further when he explained that reference frames subject to constant acceleration, such as those free-falling toward a gravitating object, were physically equivalent to inertial reference frames. This is why, for example, astronauts experience weightlessness when in free-fall orbit around the Earth, and why Newton's Laws of Motion are more easily discernible in such environments. If an astronaut places an object with mass in mid-air next to herself, it will remain stationary with respect to the astronaut due to its inertia. This is the same thing that would occur if the astronaut and the object were in intergalactic space with no net force of gravity acting on their shared reference frame. This principle of equivalence was one of the foundational underpinnings for the development of the general theory of relativity.[11]

### Newton's second law

A modern statement of Newton's second law is a vector differential equation:[12]

$\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{dt}} = \frac{\mathrm{d}(m \vec{v})}{\mathrm{dt}} = \frac{\mathrm{d}m}{\mathrm{dt}}\vec{v}+m\frac{\mathrm{d}\vec{v}}{\mathrm{dt}}$
$=\frac{\mathrm{d}m}{\mathrm{dt}}\frac{\mathrm{d}\vec{x}}{\mathrm{dt}}+m\frac{\mathrm{d}^2\vec{x}}{\mathrm{dt}^2}$

where $\vec{p}$ is the momentum of the system. The $\vec{F}$ in the equation represents the net (vector sum) force; in equilibrium there is zero net force by definition, but (balanced) forces may be present nevertheless. In contrast, the second law states an unbalanced force acting on an object will result in the object's momentum changing over time. [10]

If the mass of the object is constant, the term on the left cancels out $(\frac{\mathrm{d}m}{\mathrm{dt}}=0)$, and Newton's second law take the form

$\vec{F} = m\frac{\mathrm{d}^2\vec{x}}{\mathrm{dt}^2},$

or more famously,

$\vec{F} =m\vec{a}.$

It is sometimes called the "second most famous formula in physics".[13] Newton never explicitly stated the $\vec{F} =m\vec{a}$ formula for which he is often credited.

Newton's second law asserts the proportionality of acceleration and mass to force. Accelerations can be defined through kinematic measurements. However, while kinematics are well-described through reference frame analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass. General relativity offers an equivalence between space-time and mass, but lacking a coherent theory of quantum gravity, it is unclear as to how or whether this connection is relevant on microscales. With some justification, Newton's second law can be taken as a quantitative definition of mass by writing the law as an equality; the relative units of force and mass then are fixed.

The use of Newton's second law as a definition of force has been disparaged in some of the more rigorous textbooks,[3][14] because it is essentially a mathematical truism. The equality between the abstract idea of a "force" and the abstract idea of a "changing momentum vector" ultimately has no observational significance because one cannot be defined without simultaneously defining the other. What a "force" or "changing momentum" is must either be referred to an intuitive understanding of our direct perception, or be defined implicitly through a set of self-consistent mathematical formulas. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of "force" include Ernst Mach, Clifford Truesdell and Walter Noll.[15]

Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of planets along with the accelerations of their orbits allows scientists to calculate the gravitational forces on planets.

### Newton's third law

Newton's third law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. For any two objects (call them 1 and 2), Newton's third law states that any force that is applied to object 1 due to the action of object 2 is automatically accompanied by a force applied to object 2 due to the action of object 1.[16]

$\vec{F}_{1,2}=-\vec{F}_{2,1}.$

This law implies that forces always occur in action-reaction pairs.[10] If object 1 and object 2 are considered to be in the same system, then the net force on the system due to the interactions between objects 1 and 2 is zero since

$\vec{F}_{1,2}+\vec{F}_{\mathrm{2,1}}=0$.

This means that in a closed system of particles, there are no internal forces that are unbalanced. That is, action-reaction pairs of forces shared between any two objects in a closed system will not cause the center of mass of the system to accelerate. The constituent objects only accelerate with respect to each other, the system itself remains unaccelerated. Alternatively, if an external force acts on the system, then the center of mass will experience an acceleration proportional to the magnitude of the external force divided by the mass of the system.[3]

Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved. Using

$\vec{F}_{1,2} = \frac{\mathrm{d}\vec{p}_{1,2}}{\mathrm{dt}} = -\vec{F}_{2,1} = -\frac{\mathrm{d}\vec{p}_{2,1}}{\mathrm{dt}}$

and integrating with respect to time, the equation:

$\Delta{\vec{p}_{1,2}} = - \Delta{\vec{p}_{2,1}}$

is obtained. For a system which includes objects 1 and 2,

$\sum{\Delta{\vec{p}}}=\Delta{\vec{p}_{1,2}} + \Delta{\vec{p}_{2,1}} = 0$

which is the conservation of linear momentum.[17] Using the similar arguments, it is possible to generalizing this to a system of an arbitrary number of particles. This shows that exchanging momentum between constituent objects will not affect the net momentum of a system. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.[3]

## Descriptions

Free-body diagrams of an object on a flat surface and an inclined plane. Forces are resolved and added together to determine their magnitudes and the resultant.

Since forces are perceived as pushes or pulls, this can provide an intuitive understanding for describing forces.[4] As with other physical concepts (e.g. temperature), the intuitive understanding of forces is quantified using precise operational definitions that are consistent with direct observations and compared to a standard measurement scale. Through experimentation, it is determined that laboratory measurements of forces are fully consistent with the conceptual definition of force offered by Newtonian mechanics.

Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous. For example, if you know that two people are pulling on the same rope with known magnitudes of force but you do not know which direction either person is pulling, it is impossible to determine what the acceleration of the rope will be. The two people could be pulling against each other as in tug of war or the two people could be pulling in the same direction. In this simple one-dimensional example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other. Associating forces with vectors avoids such problems.

Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction.[4] When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the superposition principle. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. The resultant force can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram.[3]

Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the resultant.[18]

As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions.[19] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other; forces acting at ninety degrees to each other have no effect on each other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Force vectors can also be three-dimensional, with the third component at right-angles to the two other components.[3]

### Equilibria

Equilibrium occurs when the resultant force acting on an object is zero (that is, the vector sum of all forces is zero). There are two kinds of equilibrium: static equilibrium and dynamic equilibrium.

#### Static equilibrium

Static equilibrium was understood well before the invention of classical mechanics. Objects which are at rest have zero net force acting on them.[20]

The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, surface forces resist the downward force with equal upward force (called the normal force). The situation is one of zero net force and no acceleration.[4]

Pushing against an object on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.[4]

A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force" which is equal to the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.[3][4]

#### Dynamical equilibrium

Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.

Dynamical equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest to be correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. However, when this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.[9]

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamical equilibrium: when all the forces on an object balance but it still moves at a constant velocity.

A simple case of dynamical equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in a net zero force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. However, when kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.[3]

### Feynman diagrams

A Feynman diagram for the decay of a neutron into a proton. The W boson is between two vertices indicating a repulsion.

In modern particle physics, forces and the acceleration of particles are explained as the exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that "force" is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum, from Noether's theorem, can be directly derived from the symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions".[6] When particle A emits (creates) or absorbs (annihilates) particle B, a force accelerates particle A in response to the momentum of particle B, thereby conserving momentum as a whole. This description applies for all forces arising from fundamental interactions. While sophisticated mathematical descriptions are needed to predict, in full detail, the nature of such interactions, there is a conceptually simple way to describe such interactions through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines (similar to waves) and, in the case of virtual particle exchange, are absorbed at an adjacent vertex.[21]

The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and neutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.[21]

### Special relativity

In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's second law $\vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t$ remains formally valid.[22] But in order to be conserved, momentum must be redefined as:

$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$

where

v is the velocity and
c is the speed of light.

The relativistic expression relating force and acceleration for a particle with non-zero rest mass $m\,$ moving in the $x\,$ direction is:

$F_x = \gamma^3 m a_x \,$
$F_y = \gamma m a_y \,$
$F_z = \gamma m a_z \,$

where the Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$[23]

Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that γ is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed.

One can however restore the form of

$F^\mu = mA^\mu \,$

for use in relativity through the use of four-vectors. This relation is correct in relativity when Fμ is the four-force, m is the invariant mass, and Aμ is the four-acceleration.[24]

## Fundamental models

All the forces in the universe are based on four fundamental forces. The strong and weak forces act only at very short distances, and are responsible for holding certain nucleons and compound nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. All other forces are based on the existence of the four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli Exclusion Principle,[25] which does not allow atoms to pass through each other. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces and the Exclusion Principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.[3]

The development of fundamental theories for forces proceeded along the lines of unification of disparate ideas. For example, Isaac Newton unified the force responsible for objects falling at the surface of the Earth with the force responsible for the orbits of celestial mechanics in his universal theory of gravitation. Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through one consistent theory of electromagnetism. In the twentieth century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons.[26] This standard model of particle physics posits a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory subsequently confirmed by observation. The complete formulation of the standard model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations indicate that the standard model is incomplete. A grand unified theory allowing for the combination of the electroweak interaction with the strong force is held out as a possibility with candidate theories such as supersymmetry proposed to accommodate some of the outstanding unsolved problems in physics. Physicists are still attempting to develop self-consistent unification models that would combine all four fundamental interactions into a theory of everything. Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is string theory.[6]

### Gravity

An initially stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. An image was taken 20 flashes per second. During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 2/20ths it has dropped a total of 4 units; by 3/20ths, 9 units and so on.

What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as $\vec{g}$ and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.[27] This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m will experience a force:

$\vec{F} = m\vec{g}$

In free-fall, this force is unopposed and therefore the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reactions of their supports. For example, a person standing on the ground experiences zero net force, since his weight is balanced by a normal force exerted by the ground.[3]

Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's Laws of Planetary Motion.[28]

Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration due to gravity is proportional to the mass of the attracting body.[28] Combining these ideas gives a formula that relates the mass ($M_\oplus$) and the radius ($R_\oplus$) of the Earth to the gravitational acceleration:

$\vec{g}=-\frac{GM_\oplus}{{R_\oplus}^2} \hat{r}$

where the vector direction is given by $\hat{r}$, the unit vector directed outward from the center of the Earth.[10]

In this equation, a dimensional constant G is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant,[29] though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of G using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing the G could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation states that the force on a spherical object of mass m1 due to the gravitational pull of mass m2 is

$\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r}$

where r is the distance between the two objects' centers of mass and $\hat{r}$ is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.[10]

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the twentieth century. During that time, sophisticated methods of perturbation analysis[30] were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.[31]

It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however, despite some early indications, no such planet could be found. When Albert Einstein finally formulated his theory of general relativity (GR) he turned his attention to the problem of Mercury's orbit and found that his theory added a correction which could account for the discrepancy. This was the first time that Newton's Theory of Gravity had been shown to be less correct than an alternative.[32]

Since then, and so far, general relativity has been acknowledged as the theory which best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved space-time – defined as the shortest space-time path between two space-time events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of space-time can be observed and the force is inferred from the object's curved path. Thus, the straight line path in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".[3]

### Electromagnetic forces

The electrostatic force was first described in 1784 by Coulomb as a force which existed intrinsically between two charges.[33] The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the law of superposition. Coulomb's Law unifies all these observations into one succinct statement.[34]

Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force.[35] Thus the electric field anywhere in space is defined as

$\vec{E} = {\vec{F} \over{q}}$

where q is the magnitude of the hypothetical test charge.

Meanwhile, the Lorentz force of magnetism was discovered to exist between two electric currents. It has the same mathematical character as Coulomb's Law with the proviso that like currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be used to determine the magnetic force on an electric current at any point in space. In this case, the magnitude of the magnetic field was determined to be

$B = {F \over{I \ell}}$

where I is the magnitude of the hypothetical test current and $\ell$ is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. The fact that the Earth's magnetic field is aligned closely with the orientation of the Earth's axis causes compass magnets to become oriented because of the magnetic force pulling on the needle.

Through combining the definition of electric current as the time rate of change of electric charge, a rule of vector multiplication called Lorentz's Law describes the force on a charge moving in an magnetic field.[35] The connection between electricity and magnetism allows for the description of a unified electromagnetic force that acts on a charge. This force can be written as a sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field). Fully stated, this is the law:

$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

where $\vec{F}$ is the electromagnetic force, q is the magnitude of the charge of the particle, $\vec{E}$ is the electric field, $\vec{v}$ is the velocity of the particle which is crossed with the magnetic field ($\vec{B}$).

The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a succinct set of four equations. These "Maxwell Equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed which he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.[36]

However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave particles known as photons. In QED, photons are the fundamental exchange particle which described all interactions relating to electromagnetism including the electromagnetic force.[37]

It is a common misconception to ascribe the stiffness and rigidity of solid matter to the repulsion of like charges under the influence of the electromagnetic force. However, these characteristics actually result from the Pauli Exclusion Principle. Since electrons are fermions, they cannot occupy the same quantum mechanical state as other electrons. When the electrons in a material are densely packed together, there are not enough lower energy quantum mechanical states for them all, so some of them must be in higher energy states. This means that it takes energy to pack them together. While this effect is manifested macroscopically as a structural "force", it is technically only the result of the existence of a finite set of electron states.

### Nuclear forces

There are two "nuclear forces" which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force[38] is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force[39] is responsible for the decay of certain nucleons into leptons and other types of hadrons.[3]

The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD).[40] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong interaction is the most powerful of the four fundamental forces.

The strong force only acts directly upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between nucleons in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluons which form part of the virtual pi and rho mesons which classically transmit the nuclear force (see this topic for more). The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called colour confinement.

The weak force is due to the exchange of the heavy W and Z bosons. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately 1015 Kelvin. Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang.

## Non-fundamental models

Some forces can be modeled by making simplifying assumptions about the physical conditions. In such situations, idealized models can be utilized to gain physical insight.

### Normal force

Fn represents the normal force exerted on the object.

The normal force is the surface force which acts normal to the surface interface between two objects.[41] The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force of an object crashing into an immobile surface.[3]

### Friction

Friction is a surface force that opposes motion. The frictional force is directly related to the normal force which acts to keep two solid objects separated at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction.

The static friction force (Fsf) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (μsf) multiplied by the normal force (FN). In other words the magnitude of the static friction force satisfies the inequality:

$0 \le F_{\mathrm{sf}} \le \mu_{\mathrm{sf}} F_\mathrm{N}$.

The kinetic friction force (Fkf) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force is equal to

Fkf = μkfFN,

where μkf is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.[3]

### Continuum mechanics

When the drag force (Fd) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (Fg), the object reaches a state of dynamical equilibrium at terminal velocity.

Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. However, in real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:

$\frac{\vec{F}}{V} = - \vec{\nabla} P$

where V is the volume of the object in the fluid and P is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[3]

A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to

# FORCE

Forces are often described as pushes, pulls or lifts. They can be due to phenomena such as gravity, magnetism, or anything else that causes a mass to accelerate.

In physics, force is what causes a mass to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as the net force or resultant force). In an extended body, force may also cause rotation, deformation, or an increase in pressure for the body. Rotational effects are determined by the torques, while deformation and pressure are determined by the stresses that the forces create.[1][2]

Net force is mathematically equal to the time rate of change of the momentum of the body on which it acts.[3] Since momentum is a vector quantity (has both a magnitude and direction), force also is a vector quantity.

The concept of force has formed part of statics and dynamics since ancient times. Ancient contributions to statics culminated in the work of Archimedes in the 3rd century BC, which still forms part of modern physics. In contrast, Aristotle's dynamics incorporated intuitive misunderstandings of the role of force which were eventually corrected in the 17th century, culminating in the work of Isaac Newton.[2] Following the development of quantum mechanics, it is now understood that particles influence each another through fundamental interactions, and therefore the standard model of particle physics demands that everything experienced fundamentally as a "force" is actually mediated by gauge bosons. Only four fundamental interactions are known: strong, electromagnetic, weak (unified into one electroweak interaction in 1970s), and gravitational (in order of decreasing strength).[1]

#### History

Aristotle famously described a force as anything which causes an object to undergo "unnatural motion"

Aristotle and his followers believed that it was the natural state of massive objects on Earth (such as water and earth) to be motionless and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g. for heavy bodies to fall), which lead to "natural motion", and unnatural or forced motion, which required continued application of a force.[4] This theory, based on the everyday experience of how objects move (e.g. constant application of a force to keep a cart moving), had severe trouble accounting for the behavior of projectiles, such as the flight of arrows. Amongst physics theories, the late medieval idea that objects in forced motion carried an innate force of impetus influenced the work of Galileo. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the 17th century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.[5]

Isaac Newton is recognized as the first person to argue explicitly that, in general, a constant force causes a constant rate of change (time derivative) of momentum. In essence, he gave the first (and the only) mathematical definition of the quantity force itself - as the time-derivative of momentum: F = dp / dt. Newton went on to publish his Philosophiae Naturalis Principia Mathematica which used concepts of inertia, force, and conservation to describe the motion of all objects. [6][2]

Newton's next contribution to force theory was to unify the motions of heavenly bodies (which were assumed by Aristotle to be in a natural state of constant motion) with motion on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's Laws of Planetary Motion.[7] His model for the force of gravity was so powerful that it was used successfully to predict the existence of massive bodies such as Neptune before they were actually observed.[6]

In 1784 Charles Coulomb discovered the inverse square law of interaction between electric charges using a torsion balance; this was the second fundamental force.[8] The weak and strong forces were discovered in the 20th century through the development of nuclear physics.

With the development of quantum field theory and general relativity, it was realized that “force” is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in Quantum Electrodynamics (QED). Thus the currently known fundamental forces are more accurately called “fundamental interactions”.[9]

## Definitions

Free-body diagrams of an object on a flat surface and an inclined plane. Forces are resolved and added together to determine their magnitudes and the resultant.

Forces can be directly perceived as pushes or pulls; this can provide an intuitive framework for describing forces.[2] As with other physical concepts (e.g. temperature), the intuitive notion is quantified using operational definitions that are consistent with direct perception, but are more precise. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments prove the crucial properties that forces are additive vector quantities: they have magnitude and direction.[2] When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. As always with vector addition, this results in a parallelogram rule; the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram.[1]

Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the resultant.[10]

As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions.[11] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other; forces acting at ninety degrees to each other have no effect on each other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Force vectors can also be three-dimensional, with the third component at right-angles to the two other components.[1]

Equilibrium occurs when the resultant force acting on an object is zero (that is, the vector sum of all forces is zero). There are two kinds of equilibrium: static equilibrium and dynamic equilibrium.

### Static equilibrium

Static equilibrium was understood well before the invention of classical mechanics. Objects which are at rest have zero net force acting on them.[12]

The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, any object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, surface forces resist the downward force with equal upward force (called the normal force) and result in the object having a non-zero weight. The situation is one of zero net force and no acceleration.[2]

Pushing against an object on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.[2]

A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force" which is equal to the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.[1][2]

### Dynamical equilibrium

Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.

Dynamical equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that massive objects naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest to be correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved underneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. However, when this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.[5]

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamical equilibrium: when all the forces on an object balance but it still moves at a constant velocity.

A simple case of dynamical equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in a net zero force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. However, when kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.[1]

### Newtonian definitions

Though Sir Isaac Newton's most famous equation is F=ma, he actually wrote down a different form for his second law of motion that used differential calculus.

In Principia Mathematica, Newton set out three laws of motion which have direct relevance to the way forces are described in physics.[6]

#### Newton's first law

Newton's first law of motion sets forth the conditions required for equilibrium and effectively defines the inertia that can be related to the mass of an object. Taking the Aristotelian idea of "natural states", the condition of constant velocity whether it be zero or nonzero is now considered the "natural state" of all massive objects. Objects will continue to move in a state of constant velocity unless acted upon by an unbalanced external force.[6]

#### Newton's second law

Force is often defined using Newton's second law, as the product of mass m multiplied by acceleration $\vec{a}$:

$\vec{F} =m\vec{a},$

sometimes called the "second most famous formula in physics". Newton never stated explicitly the F=ma formula for which he is often credited. Newton's second law is described in his Principia Mathematica as a vector differential equation:

$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m \vec{v})}{dt}$

where $\vec{p}$ is the momentum of the system.[6] Force is the rate of change of momentum over time and momentum or acceleration is the rate of change of velocity over time.

The use of Newton's second law in either of these forms as a definition of force is generally disparaged in more rigorous textbooks,[1] [13] because this removes all empirical content from the law. In fact, the $\vec{F}$ in this equation represents the net (vector sum) force; in equilibrium this is zero by definition, but (balanced) forces are present nevertheless. Instead, Newton's law is meaningful because it asserts the proportionality of two quantities which can be defined without reference to it. Thus, the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity (therefore zero acceleration) is objectively wrong and not just a consequence of a poor choice of definition. With rather more justification, Newton's second law can be taken as a quantitative definition of mass by writing the law as an equality, the relative units of force and mass are fixed.

Given the empirical success of Newton's law, it is sometimes used to measure the strength of forces (for instance, using astronomical orbits to determine gravitational forces). Nevertheless, the force and the quantities used to measure it remain distinct concepts.

The definition of force is sometimes regarded as problematic, since it must either ultimately be referred to our intuitive understanding of our direct perceptions, or be defined implicitly through a set of self-consistent mathematical formulae. Notable physicists, philosophers and mathematicians who have sought a more explicit definition include Ernst Mach, Clifford Truesdell and Walter Noll.[14]

#### Newton's third law

Newton's third law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. For any two objects (call them 1 and 2), Newton's third law states that

$\vec{F}_{\mathrm{1 on 2}}=-\vec{F}_{\mathrm{2 on 1}}.$

This law implies that forces always occur in action-reaction pairs.[6] Any force that is applied to object 1 due to the action of object 2 is automatically accompanied by a force applied to object 2 due to the action of object 1. If object 1 and object 2 are considered to be in the same system, then the net force on the system due to the interactions between objects 1 and 2 is zero since

$\vec{F}_{\mathrm{1 on 2}}+\vec{F}_{\mathrm{2 on 1}}=0$.

This means that systems cannot create internal forces that are unbalanced. However, if objects 1 and 2 are considered to be in separate systems, then the two systems will each experience an unbalanced force and accelerate with respect to each other according to Newton's second law.[1]

Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved. Using

$\vec{F}_{\mathrm{1 on 2}} = \frac{d\vec{p}_{\mathrm{1 on 2}}}{dt} = -\vec{F}_{\mathrm{2 on 1}} = -\frac{d\vec{p}_{\mathrm{2 on 1}}}{dt}$

and integrating with respect to time, the equation:

$\Delta{\vec{p}_{\mathrm{1 on 2}}} = - \Delta{\vec{p}_{\mathrm{2 on 1}}}$

is obtained. For a system which includes objects 1 and 2,

$\sum{\Delta{\vec{p}}}=\Delta{\vec{p}_{\mathrm{1 on 2}}} + \Delta{\vec{p}_{\mathrm{2 on 1}}} = 0$

which is the conservation of linear momentum. Generalizing this to a system of an arbitrary number of particles is straightforward. This shows that exchanging momentum between constituent objects will not affect the net momentum of a system. In general, as long as all forces are due to the interaction of massive objects, it is possible to define a system such that net momentum is never lost nor gained.[1]

## Fundamental models

All the forces in the Universe are all based on four fundamental forces. The strong and weak forces act only at very short distances, and are responsible for holding certain nucleons and compound nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. All other forces are based on the existence of the four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli Exclusion Principle, which does not allow atoms to pass through each other. The forces in springs modeled by Hooke's law are also the result of electromagnetic forces and the Exclusion Principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.[1]

The modern quantum mechanical view of the first three fundamental forces (all except gravity) is that particles of matter (fermions) do not directly interact with each other but rather by exchange of virtual particles (bosons). This exchange results in what we call electromagnetic interaction (Coulomb force is one example of electromagnetic interaction).

It is a common misconception to ascribe the stiffness and rigidity of solid matter to the repulsion of like charges under the influence of the electromagnetic force. However, these characteristics actually result from the Pauli Exclusion Principle. Since electrons are fermions, the rules of quantum mechanics forbid them to occupy the same quantum mechanical state as other electrons. When the electrons in a material are densely packed together, there are not enough lower energy quantum mechanical states for them all, so some of them must be in higher energy states. This means that it takes energy to pack them together. While this effect is manifested macroscopically as a structural "force", it is technically only the result of the existence of a finite ensemble of electron orbitals rather than being dependent directly on, for example, the electromagnetic force. All the electromagnetic force does is mediate the particular distribution and variety of the orbitals for the collection of atoms. Even if the electromagnetic force did not exist, fermions would still be excluded from occupying the same state. Such degeneracy pressure forces can only be overcome when there is enough energy in the system so that other fundamental interactions between the particles can take precedence (for example, electrons and protons interacting via the weak nuclear force to become neutrons). However, this kind of event only occurs in very extreme circumstances such as particle accelerators or neutron stars.[1]

### Gravity

An initially stationary object is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. An image was taken 20 flashes per second. During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 2/20ths it has dropped at total of 4 units; by 3/20ths, 9 units and so on.

What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every body in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity at the surface of the Earth is usually designated as $\vec{g}$ and has a value of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m will experience a force:

$\vec{F} = m\vec{g}$

In free-fall, this force is unopposed and therefore the net force on the object is the force of gravity. For objects not in free-fall, the force of gravity is opposed by the weight of the object. For example, a person standing on the ground experiences zero net force, since the force of gravity is balanced by the weight of the person that is manifested by a normal force exerted on the person by the ground.[1]

Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the mass of the gravitating object directly affected the acceleration due to gravity.[7] Combining these ideas gives a formula that relates the mass of the Earth ($M_\oplus$), the radius of the Earth ($R_\oplus$) to the acceleration due to gravity:

$\vec{g}=-\frac{GM_\oplus}{{R_\oplus}^2} \hat{r}$

where the vector direction is given by $\hat{r}$ which is the unit vector directed outward from the center of the Earth.[6]

In this equation, a dimensional constant G is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant, though it was of an unknown value in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of G using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing the G could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation between two massive bodies is

$\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r}$

where m1 is the mass of first object and m2 is the mass of the second object.[6]

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the twentieth century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. These techniques are so powerful that they can be used to predict precisely the motion of celestial bodies to an arbitrary precision at any length of time in the future. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.[15]

It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however, despite some early indications, no such planet could be found. When Albert Einstein finally formulated his theory of general relativity (GR) he turned his attention to the problem of Mercury's orbit and found that his theory added a correction which could account for the discrepancy. This was the first time that Newton's Theory of Gravity had been shown to be less correct than an alternative.[16]

Since then, and so far, general relativity has been acknowledged as the theory which best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved space-time – defined as the shortest space-time path between two space-time events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of space-time can be observed and the force is inferred from the object's curved path. Thus, the straight line path in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the body is what we label as "gravitational force".[1]

### Electromagnetic forces

The electrostatic force was first described in 1784 by Coulomb as a force which existed intrinsically between two charges. The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the law of superposition. Unifying all these observations into one succinct statement became known as Coulomb's Law.[8]

As the mathematical formalism for Coulomb's Law, physicists of the eighteenth and nineteenth century became interested in the electric field which could be used to determine the electrostatic force on an electric charge at any point in space. Knowing the magnitude of the electric field automatically gave the information required to know what the force applied on a given electric charge would be.[1]

Meanwhile, knowledge of the Lorentz force of magnetism was developing. This force can be described as a force that exists between two electric currents. It has the same mathematical character as Coulomb's Law with the proviso that like currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be defined which can be used to determine the magnetic force on an electric current at any point in space. Combining the definition of electric current as the time rate of change of electric charge yields a law of vector multiplication called Lorentz's Law which determines the force on a moving charge provided with an electric field.[1]

Thus a full theory of the electromagnetic force on a charge can be written as a sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field). Fully stated, this is the law:

$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

where $\vec{F}$ is the electromagnetic force, q is the magnitude of the charge of the particle, $\vec{v}$ is the velocity of the particle, $\vec{E}$ is the electric field, and $\vec{B}$ is the magnetic field.

The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a succinct set of four equations. These "Maxwell Equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed which he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.[17]

However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave particles known as photons. In QED, photons are the fundamental exchange particle which described all interactions relating to electromagnetism including the electromagnetic force.[18]

### Nuclear forces

There are two "nuclear forces" which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force is responsible for the decay of certain nucleons into leptons and other types of hadrons.[1]

The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD)[19]. The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong interaction is the most powerful of the four fundamental forces.

The strong force only acts directly upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between nucleons in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluons which form part of the virtual pi and rho mesons which classically transmit the nuclear force (see this topic for more). The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called colour confinement.

The weak force is due to the exchange of the heavy W and Z bosons. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately 1015 Kelvin. Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang.

## Non-fundamental models

Some forces are be modeled by making simplifying assumptions about the physical conditions. In such situations, idealized models can be utilized to gain physical insight.

### Normal force

The normal force is the surface force which acts normal to the surface interface between two objects. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object.[1]

### Friction

Friction is a surface force that opposes motion. The frictional force is directly related to the normal force which acts to keep two solid objects separated at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction.[1]

The static friction force (Fsf) will exactly oppose forces applied to a body parallel to a surface contact up to the limit specified by the coefficient of static friction (μsf) multiplied by the normal force (FN). In other words the magnitude of the static friction force satisfies the inequality:

$0 \le F_{sf} \le \mu_{sf} F_N$.

The kinetic friction force (Fkf) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force is equal to

Fkf = μkfFN,

where μkf is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.

### Continuum mechanics

When the drag force (Fd) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (Fg), the object reaches a state of dynamical equilibrium at terminal velocity.

When fluid flows, differences in pressure result in forces being directed along the pressure gradients as follows:

$\frac{\vec{F}}{V} = - \vec{\nabla} P$

where V is the volume of the object in the fluid and P is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[1]

A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes's drag" the force is approximately proportional to the velocity, but opposite in direction:

$\vec{F}_d = - b \vec{v} \,$

where:

b is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
$\vec{v}$ is the velocity of the object.[1]

More formally, forces in continuum mechanics are fully desribed by a stress tensor with terms that are rougly defined as

$\sigma = \frac{F}{A}$

where A is the relevant cross-sectional area for the volume for which the stress-tensor is being calculated. This formalism includes pressure terms which act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms which act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause deformation including also tensile stresses and compressions.

### Tension

Tension forces can be idealized using ideal strings which are massless, frictionless, unbreakable, and unstretchable. They can be combined with ideal pulleys which are essentially devices that can allow for ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action-reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.[20] By connecting the same string multiple times to the same object, the tension force can be multiplied. In order to do this, a system of pulleys must be employed. While such set-ups allow for an increase in force the corresponding increase in distance associated with the greater length of string conserves energy as all simple machines must.[21][1]

### Hooke's Law

Main article: Hooke's Law
Fk is the Hooke's law force that responds to the load on the end of the spring.

Ideal springs that are massless, frictionless, unbreakable, and infinitely stretchable can be used to model elastic forces. Such springs exert forces that push when contracted or pull when extended in proportion to the displacement of the spring from its equilibrium size. If Δx is the displacement, the force exerted by the spring is equal to:

$\vec{F}=-k \Delta \vec{x}$

where k is the Hooke's Law constant that is different for every spring.[1][22]

### Centripetal force

For an object accelerating in circular motion, the unbalanced force acting on the object is equal to[23]

$\vec{F} = - \frac{mv^2 \hat{r}}{r}$

where m is the mass of the object, v is the velocity of the object and r is the distance to the center of the circular path and $\hat{r}$ is the unit vector pointing in the radial direction outwards from the center. This means that the unbalanced centripetal force felt by any object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. The unbalanced force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force which accelerates the object by either slowing it down or speeding it up and the radial (centripetal) force which changes its direction.[1]

### Fictitious forces

There are forces which are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force.[24] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[1]

## Rotations and torque

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system.

Forces that cause extended objects to rotate are associated with torques. Mathematically, torque is defined as the cross-product:

$\vec{\tau} = \vec{r} \times \vec{F}$

where

$\vec{r}$ is the particle's position vector relative to a pivot
$\vec{F}$ is the force acting on the particle.

Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. All the formal treatments of Newton's Laws that applied to forces equivalently apply to torques. Thus, as a consequence of Newton's First Law of Motion, there exists rotational inertia that ensures that all bodies rotating at a constant angular velocity maintain that angular velocity unless acted upon by unbalanced torques. Likewise, Newton's Second Law of Motion can be used to derive an alternative definition of torque:

$\vec{\tau} = I \times \vec{\alpha}$

where

I is the moment of inertia of the particle
$\vec{\alpha}$ is the angular acceleration of the particle.

This provides a definition for the moment of inertia which is the rotational equivalent for mass. In more advanced treatments of mechanics, the moment of inertia acts as a tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.

Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:

$\vec{\tau} = \frac{d\vec{L}}{dt}$[25]

where $\vec{L}$ is the angular momentum of the particle.

Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques[26], and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.

## Kinematic integrals

Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse:

$\vec{I}=\int{\vec{F} dt}$

which, by Newton's Second Law, must be equivalent to the change in momentum (yielding the Impulse momentum theorem).

Similarly, integrating with respect to position gives a definition for the work done by a force:

$W=\int{\vec{F} \cdot{d\vec{x}}}$

which, in a system where all the forces are conservative (see below) is equivalent to changes in kinetic and potential energy (yielding the Work energy theorem). The time derivative of the definition of work gives a definition for power in term of force and the velocity ($\vec{v}$):

$P=\frac{dW}{dt}=\int{\vec{F} \cdot{d\vec{v}}}$

## Potential energy

Instead of a force, the mathematically equivalent concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. Restating mathematically the definition of energy (via definition of work), a potential scalar field $U(\vec{r})$ is defined as that field whose gradient is equal and opposite to the force produced at every point:

$\vec{F}=-\vec{\nabla} U.$

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential.[1]

### Conservative forces

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space[27], and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.[1]

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces have models which are dependent on a position often given as a radial vector $\vec{r}$ emanating from spherically symmetric potentials.[28] Examples of this follow:

For gravity:

$\vec{F} = - \frac{G m_1 m_2 \vec{r}}{r^3}$

where G is the gravitational constant, and mn is the mass of object n.

For electrostatic forces:

$\vec{F} = \frac{q_{1} q_{2} \vec{r}}{4 \pi \epsilon_{0} r^3}$

where ε0 is electric permittivity of free space, and qn is the electric charge of object n.

For spring forces:

$\vec{F} = - k \vec{r}$

where k is the spring constant.[1]

### Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations which yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model which is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.[1]

The connection between macroscopic non-conservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second Law of Thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.[1]

## Units of measurement

The SI unit used to measure force is the Newton (symbol N), which is equivalent to kilogram meter per second squared (kg·m·s−2). The CGS unit is the dyne (g·cm·s−2).

The foot-pound-second Imperial unit of force is the pound-force, defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2 at the Earth's surface. In some fields, such as structural engineering, the term 'pound' may refer instead to a unit of force; in this case the pound force is defined as the force that will accelerate a mass of one slug by one foot per second squared. An alternate unit in the same system is the poundal, defined as the force required to accelerate a one pound mass at a rate of one foot per second squared. The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law.

The pound-force has a metric counterpart, less commonly used than the Newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass which accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated; however it is still see use for some purposes as engine thrust, bicycle spoke tension, torque wrench settings and engine torque output.

# Force

In physics, force is action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force). In an extended body, force may also cause rotation, deformation, or an increase in pressure for the body. Rotational effects are determined by the torques, while deformation and pressure are determined by the stresses that the forces create.

Net force is mathematically equal to the rate of change of the momentum of the body on which it acts. Since momentum is a vector quantity (has both a magnitude and direction), force also is a vector quantity.

The concept of force has formed part of statics and dynamics since ancient times. Ancient contributions to statics culminated in the work of Archimedes in the 3rd century BC, which still forms part of modern physics. In contrast, Aristotle's dynamics incorporated intuitive misunderstandings of the role of force which were eventually corrected in the 17th century, culminating in the work of Isaac Newton. Following the development of quantum mechanics it is now understood that particles influence each another through fundamental interactions, making force a useful concept only on the macroscopic level. Only four fundamental interactions are known: strong, electromagnetic, weak (unified into one electroweak interaction in 1970s), and gravitational (in order of decreasing strength).

## History

Aristotle and his followers believed that it was the natural state of objects on Earth to be motionless and that they tended towards that state if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g. for heavy bodies to fall), which lead to "natural motion", and unnatural or forced motion, which required continued application of a force. But this theory, although based on the everyday experience of how objects move (e.g. a horse and cart), had severe trouble accounting for projectiles, such as the flight of arrows. Several theories were discussed over the centuries, and the late medieval idea that objects in forced motion carried an innate force of impetus was influential on the work of Galileo. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the 17th century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force - usually friction.

Isaac Newton is recognised as having argued explicitly for the first time that, in general, a constant force causes a constant rate of change (time derivative) of momentum. In essense, he gave the first (and the only) mathematical definition of the quantity force itself - as being the time-derivative of momentum: F = dp/dt.

In 1784 Charles Coulomb discovered the inverse square law of interaction between electric charges using a torsion balance, which was the second fundamental force. The weak and strong forces were discovered in the 20th century.

With the development of quantum field theory and general relativity it was realized that “force” is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in QED). Thus currently known fundamental forces are more accurately called “fundamental interactions”.

## Types of force

Although there are apparently many types of forces in the Universe, they are all based on four fundamental forces. The strong and weak forces only act at very short distances and are responsible for holding certain nucleons and compound nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. The Pauli exclusion principle is responsible for the tendency of atoms not to overlap each other, and is thus responsible for the "stiffness" or "rigidness" of matter, but this also depends on the electromagnetic force which binds the constituents of every atom.

All other forces are based on these four. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli exclusion principle, which does not allow atoms to pass through each other. The forces in springs modeled by Hooke's law are also the result of electromagnetic forces and the exclusion principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.

There is currently some debate to whether there are five forces not four, due to the discovery of dark energy, which could be just an energy of vacuum fluctuations, or it could be a new type of energy resulting in a repulsive force.

The modern quantum mechanical view of the first three fundamental forces (all except gravity) is that particles of matter (fermions) do not directly interact with each other but rather by exchange of virtual particles (bosons). This exchange results in what we call electromagnetic interaction (Coulomb force is one example of electromagnetic interaction).

In general relativity, gravitation is not viewed as a force. Rather, objects moving freely in gravitational fields simply undergo inertial motion along a straight line in the curved space-time - defined as the shortest space-time path between two space-time points. This straight line in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola shape as it is in a uniform gravitational field. Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the body is what we label as "gravitational force".

## Examples

• A heavy object is in free fall. Its momentum changes as dp/dt = mdv/dt = ma =mg (if the mass m is constant), thus we call the quantity mg a "gravitational force" acting on the object. This is the definition of weight (w=mg) of an object.
• A heavy object on a table is pulled (attracted) downward toward the floor by the force of gravity (i.e., its weight). At the same time, the table resists the downward force with equal upward force (called the normal force), resulting in zero net force, and no acceleration. (If the object is a person, he actually feels the normal force acting on him from below.)
• A heavy object on a table is gently pushed in a sideways direction by a finger. However, it doesn't move because the force of the finger on the object is now opposed by a new force of static friction, generated between the object and the table surface. This newly generated force exactly balances the force exerted on the object by the finger, and again no acceleration occurs. The static friction increases or decreases automatically. If the force of the finger is increased (up to a point), the opposing sideways force of static friction increases exactly to the point of perfect opposition.
• A heavy object on a table is pushed by a finger hard enough that static friction cannot generate sufficient force to match the force exerted by the finger, and the object starts sliding across the surface. If the finger is moved with a constant velocity, it needs to apply a force that exactly cancels the force of kinetic friction from the surface of the table and then the object moves with the same constant velocity. Here it seems to the naive observer that application of a force produces a velocity (rather than an acceleration). However, the velocity is constant only because the force of the finger and the kinetic friction cancel each other. Without friction, the object would continually accelerate in response to a constant force.
• A heavy object reaches the edge of the table and falls. Now the object, subjected to the constant force of its weight, but freed of the normal force and friction forces from the table, gains in velocity in direct proportion to the time of fall, and thus (before it reaches velocities where air resistance forces becomes significant compared to gravity forces) its rate of gain in momentum and velocity is constant. These facts were first discovered by Galileo.
• A heavy object is suspended on a spring scale. Because object is not moving (so time derivative of its momentum is zero) then along with acceleration of free fall g it has to experience equal and oppositely directed acceleration a=-g due to the action of the spring. This acceleration multiplied by the mass of the object is what we label as "spring reaction force" which is obviousely equal and opposite to object's weight mg. Knowing the mass (say, 1 kg) and the acceleration of free fall (say, 9.80 m/s^2) we can calibrate spring scale by a mark "9.8 N". Attaching various masses (2 kg, 3 kg ...) we can calibrate spring scale and then use this calibrated scale to measure many other forces (friction, reaction forces, elecric force, magnetic force, etc).

## Quantitative definition

We have an intuitive grasp of the notion of force, since forces can be directly perceived as a push or pull. As with other physical concepts (e.g. temperature), the intuitive notion is quantified using operational definitions that are consistent with direct perception, but more precise. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces cancelled each other out. Such experiments prove the crucial properties that forces are additive vector quantities: they have magnitude and direction. So, when two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. As with all vector addition this results in a parallelogram rule: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram.

As well as being added, forces can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.

The simplest case of static equilibrium is when two forces are equal in magnitude but opposite in direction. This remains the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. Using such tools, several quantitative force laws were discovered: that the force of gravity is proportional to volume for objects made of a given material (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs: all these were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion.

Force is sometimes defined using Newton's second law, as the product of mass m times acceleration $\vec{a}$:

$\vec{F} = m \vec{a}$

(or, more generally, as the rate of change of momentum). This approach is disparaged by the large majority of textbooks. By making this a definition of force, all empirical content is removed from the law. In fact, the $\vec{F}$ in this equation represents the net (vector sum) force; in static equilibrium this is zero by definition, but (balanced) forces are present nevertheless. Instead, Newton's law is meaningful because it asserts the proportionality of two quantities which can be defined without reference to it. Thus, the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity (therefore zero acceleration) is objectively wrong and not just a consequence of a poor choice definition. With rather more justification, Newton's second law can be taken as a quantitative definition of mass; certainly, by writing the law as an equality, the relative units of force and mass are fixed.

Given the empirical success of Newton's law, it is sometimes used to measure the strength of forces (for instance, using astronomical orbits to determine gravitational forces). Nevertheless, the force and the (mass times acceleration) used to measure it remain distinct concepts.

The definition of force is sometimes regarded as problematic, since it must either ultimately be referred to our intuitive understanding of our direct perceptions, or be defined implicitly through a set of self-consistent mathematical formulae. Notable physicists, philosophers and mathematicians who have sought a more explicit definition include Ernst Mach, Clifford Truesdell and Walter Noll.[2]

## Force in special relativity

In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate a body). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires a greater force to accelerate it the same amount than it did at a lower velocity. The definition $\vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t$ remains valid. But in order to be conserved, momentum must be redefined as:

$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$

where

v is the velocity and
c is the speed of light.

The relativistic expression relating force and acceleration for a particle with non-zero rest mass $m\,$ moving in the $x\,$ direction is:

$F_x = \gamma^3 m a_x \,$
$F_y = \gamma m a_y \,$
$F_z = \gamma m a_z \,$

where the Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that γ is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed.

One can however restore the form of

$F^\mu = mA^\mu \,$

for use in relativity through the use of four-vectors. This relation is correct in relativity when Fμ is the four-force, m is the invariant mass, and Aμ is the four-acceleration.

## Force and potential

Instead of a force, the mathematically equivalent concept of a potential energy field can be used for convenience. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. Restating mathematically the definition of energy (via definition of work), a potential scalar field $U(\vec{r})$ is defined as that field whose gradient is equal and opposite to the force produced at every point:

$\vec{F}=-\vec{\nabla} U$

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential.

### Conservative forces

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces, therefore, have models which are dependent on a position often given as a radial vector $\vec{r}$ eminating from spherically symmetric potentials. Examples of this follow:

For gravity:

$\vec{F} = - \frac{G m_1 m_2 \vec{r}}{r^3}$

where G is the gravitational constant, mn is the mass of object n.

For electrostatic forces:

$\vec{F} = - \frac{q_{1} q_{2} \vec{r}}{4 \pi \epsilon_{0} r^3}$

where ε0 is electric permittivity of free space, qn is the electric charge of object n.

For spring forces:

$\vec{F} = - k \vec{r}$

where k is the spring constant.

### Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations which yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model which is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.

The connection between macroscopic non-conservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system and are often associated with the transfer of heat. According to the Second Law of Thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.

## Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s−2. The earlier CGS unit is the dyne. The relationship F=m·a can be used with either of these. In Imperial engineering units, if F is measured in "pounds force" or "lbf", and a in feet per second squared, then m must be measured in slugs. Similarly, if mass is measured in pounds mass, and a in feet per second squared, the force must be measured in poundals. The units of slugs and poundals are specifically designed to avoid a constant of proportionality in this equation.

A more general form F=k·m·a is needed if consistent units are not used. Here, the constant k is a conversion factor dependent upon the units being used.

When the standard g (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf.

The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard g which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity.

By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the hyl, TME (from a German acronym), and mug (from metric slug).

Another unit of force called the poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl. The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern SI system, but is still used in applications such as:

• Thrust of jet and rocket engines
• Spoke tension of bicycles
• Draw weight of bows
• Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
• Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
• Pressure gauges in "kg/cm²" or "kgf/cm²"

In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force.

The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to distinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.

### Conversions

Below are several conversion factors between various measurements of force:

• 1 dyne = 10-5 newtons
• 1 kgf (kilopond kp) = 9.80665 newtons
• 1 metric slug = 9.80665 kg
• 1 lbf = 32.174 poundals
• 1 slug = 32.174 lb
• 1 kgf = 2.2046 lbf

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