# Force

{{#invoke:Hatnote|hatnote}} Template:Infobox Physical quantity

Template:Classical mechanics In physics, a force is any interaction which tends to change the motion of an object.[1] In other words, a force can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate. Force can also be described by intuitive concepts such as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object. As a formula, this is expressed as:

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

where the arrows imply a vector quantity possessing both magnitude and direction.

Related concepts to force include: thrust, which increases the velocity of an object; drag, which decreases the velocity of an object; and torque which produces changes in rotational speed of an object. In an extended body, each part usually applies forces on the adjacent parts; the distribution of such forces through the body is the so-called mechanical stress. Pressure is a simple type of stress. Stress usually causes deformation of solid materials, or flow in fluids.

## Development of the concept

Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes 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.[2] A fundamental error was the belief that a force is required 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 were not improved-on for nearly three hundred years.[3] By the early 20th century, Einstein developed a theory 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 a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged 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.[4]:2–10[5]:79 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.[6]

## Pre-Newtonian concepts

Aristotle famously described a force as anything that 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 for the same amount of work. 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.[2]

Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground and that they will stay that way 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 the archer moves the projectile was at the start of the flight, and while the projectile sailed through the air, no discernible efficient cause acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation demands a continuum like air for change of place in general.[8]

Aristotelian physics began facing criticism in Medieval science, first by John Philoponus in the 6th century.

The shortcomings of Aristotelian physics would not be fully corrected until the 17th 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 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.[9]

## Newtonian mechanics

{{#invoke:main|main}} Sir Isaac Newton sought to describe the motion of all objects using the concepts of inertia and force, and in doing so he found that they obey certain conservation laws. In 1687, Newton went on to publish his thesis Philosophiæ Naturalis Principia Mathematica.[3][10] In this work Newton set out three laws of motion that to this day are the way forces are described in physics.[10]

### First law

{{#invoke:main|main}} 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 instance, 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 everything 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 himself, 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]

Though Sir Isaac Newton's most famous equation is
${\displaystyle \scriptstyle {{\vec {F}}=m{\vec {a}}}}$, he actually wrote down a different form for his second law of motion that did not use differential calculus.

### Second law

{{#invoke:main|main}} A modern statement of Newton's Second Law is a vector differential equation:[Note 1]

${\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}$

where ${\displaystyle \scriptstyle {\vec {p}}}$ is the momentum of the system, and ${\displaystyle \scriptstyle {\vec {F}}}$ is 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]

By the definition of momentum,

${\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\frac {\mathrm {d} \left(m{\vec {v}}\right)}{\mathrm {d} t}},}$

where m is the mass and ${\displaystyle \scriptstyle {\vec {v}}}$ is the velocity.[4]:9-1,9-2

Newton's second law applies only to a system of constant mass,[Note 2] and hence m may be moved outside the derivative operator. The equation then becomes

${\displaystyle {\vec {F}}=m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}.}$

By substituting the definition of acceleration, the algebraic version of Newton's Second Law is derived:

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

Newton never explicitly stated the formula in the reduced form above.[12]

Newton's Second Law asserts the direct proportionality of acceleration to force and the inverse proportionality of acceleration to mass. 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.

### Nuclear

{{#invoke:main|main}} {{#invoke:see also|seealso}} 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[18]:940 is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force[18]:951 is responsible for the decay of certain nucleons into leptons and other types of hadrons.[4][5]

The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD).[38] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The (aptly named) strong interaction is the "strongest" 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 color 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 kelvins. 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 forces

Some forces are consequences of the fundamental ones. In such situations, idealized models can be utilized to gain physical insight.

### Normal force

FN represents the normal force exerted on the object.

{{#invoke:main|main}} The normal force is due to repulsive forces of interaction between atoms at close contact. When their electron clouds overlap, Pauli repulsion (due to fermionic nature of electrons) follows resulting in the force which acts in a direction normal to the surface interface between two objects.[18]:93 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 on an object crashing into an immobile surface.[4][5]

### Friction

{{#invoke:main|main}} Friction is a surface force that opposes relative 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 (${\displaystyle F_{\mathrm {sf} }}$) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (${\displaystyle \mu _{\mathrm {sf} }}$) multiplied by the normal force (${\displaystyle F_{N}}$). In other words the magnitude of the static friction force satisfies the inequality:

${\displaystyle 0\leq F_{\mathrm {sf} }\leq \mu _{\mathrm {sf} }F_{\mathrm {N} }}$.

The kinetic friction force (${\displaystyle F_{\mathrm {kf} }}$) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:

${\displaystyle F_{\mathrm {kf} }=\mu _{\mathrm {kf} }F_{\mathrm {N} }}$,

where ${\displaystyle \mu _{\mathrm {kf} }}$ is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.

### Tension

{{#invoke:main|main}} Tension forces can be modeled using ideal strings which are massless, frictionless, unbreakable, and unstretchable. They can be combined with ideal pulleys which allow 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.[39] By connecting the same string multiple times to the same object through the use of a set-up that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. However, even though such machines allow for an increase in force, there is a corresponding increase in the length of string that must be displaced in order to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.[4][5][40]

### Elastic force

{{#invoke:main|main}}

Fk is the force that responds to the load on the spring

An elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position.[41] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If ${\displaystyle \Delta x}$ is the displacement, the force exerted by an ideal spring equals:

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

where ${\displaystyle k}$ is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.[4][5]

### Continuum mechanics

When the drag force (${\displaystyle F_{d}}$) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (${\displaystyle F_{g}}$), the object reaches a state of dynamic equilibrium at terminal velocity.

{{#invoke:main|main}} 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:

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

where ${\displaystyle V}$ is the volume of the object in the fluid and ${\displaystyle P}$ 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.[4][5]

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 the velocity, but opposite in direction:

${\displaystyle {\vec {F}}_{\mathrm {d} }=-b{\vec {v}}\,}$

where:

${\displaystyle b}$ is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
${\displaystyle \scriptstyle {\vec {v}}}$ is the velocity of the object.[4][5]

More formally, forces in continuum mechanics are fully described by a stresstensor with terms that are roughly defined as

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

where ${\displaystyle A}$ is the relevant cross-sectional area for the volume for which the stress-tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.[3][5]:133–134[35]:38-1–38-11

### Fictitious forces

{{#invoke:main|main}} 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.[42] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[4][5] Because these forces are not genuine they are also referred to as "pseudo forces".[4]:12-11

In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry. As an extension, Kaluza–Klein theory and string theory ascribe electromagnetism and the other fundamental forces respectively to the curvature of differently scaled dimensions, which would ultimately imply that all forces are fictitious.

## Rotations and torque

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

{{#invoke:main|main}} Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force ${\displaystyle \scriptstyle {\vec {F}}}$ is defined relative to an arbitrary reference point as the cross-product:

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

where

${\displaystyle \scriptstyle {\vec {r}}}$ is the position vector of the force application point relative to the reference point.

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. As a consequence of Newton's First Law of Motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's Second Law of Motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:

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

where

${\displaystyle I}$ is the moment of inertia of the body
${\displaystyle \scriptstyle {\vec {\alpha }}}$ is the angular acceleration of the body.

This provides a definition for the moment of inertia which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the 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:

${\displaystyle {\vec {\tau }}={\frac {\mathrm {d} {\vec {L}}}{\mathrm {dt} }},}$[43] where ${\displaystyle \scriptstyle {\vec {L}}}$ 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,[44] and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.

### Centripetal force

{{#invoke:main|main}} For an object accelerating in circular motion, the unbalanced force acting on the object equals:[45]

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

where ${\displaystyle m}$ is the mass of the object, ${\displaystyle v}$ is the velocity of the object and ${\displaystyle r}$ is the distance to the center of the circular path and ${\displaystyle \scriptstyle {\hat {r}}}$ 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.[4][5]

## Kinematic integrals

{{#invoke:main|main}} 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:[46]

${\displaystyle {\vec {I}}=\int _{t_{1}}^{t_{2}}{{\vec {F}}\mathrm {d} t}}$

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:[4]:13-3

${\displaystyle W=\int _{{\vec {x}}_{1}}^{{\vec {x}}_{2}}{{\vec {F}}\cdot {\mathrm {d} {\vec {x}}}}}$

which is equivalent to changes in kinetic energy (yielding the work energy theorem).[4]:13-3

Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change ${\displaystyle \scriptstyle {d}{\vec {x}}}$ in a time interval dt:[4]:13-2

${\displaystyle {\text{d}}W\,=\,{\frac {{\text{d}}W}{{\text{d}}{\vec {x}}}}\,\cdot \,{\text{d}}{\vec {x}}\,=\,{\vec {F}}\,\cdot \,{\text{d}}{\vec {x}},\qquad {\text{ so }}\quad P\,=\,{\frac {{\text{d}}W}{{\text{d}}t}}\,=\,{\frac {{\text{d}}W}{{\text{d}}{\vec {x}}}}\,\cdot \,{\frac {{\text{d}}{\vec {x}}}{{\text{d}}t}}\,=\,{\vec {F}}\,\cdot \,{\vec {v}},}$

## Potential energy

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

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

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.[4][5]

### Conservative forces

{{#invoke:main|main}} 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,[47] 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.[4][5]

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models which are dependent on a position often given as a radial vector ${\displaystyle \scriptstyle {\vec {r}}}$ emanating from spherically symmetric potentials.[48] Examples of this follow:

For gravity:

${\displaystyle {\vec {F}}=-{\frac {Gm_{1}m_{2}{\vec {r}}}{r^{3}}}}$

where ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle m_{n}}$ is the mass of object n.

For electrostatic forces:

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

For spring forces:

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

### 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.[4][5]

The connection between macroscopic nonconservative 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.[4][5]

## Units of measurement

The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.[49] The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes.

The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2.[49] The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force.[49]

An alternative unit of force in a different foot-pound-second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one pound mass at a rate of one foot per second squared.[49] 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.[49] 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 still sees use for some purposes as expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. Other arcane units of force include the sthène which is equivalent to 1000 N and the kip which is equivalent to 1000 lbf.

Units of force
Template:Navbar newton
(SI unit)
dyne kilogram-force,
kilopond
pound-force poundal
1 N ≡ 1 kg·m/s2 = 105 dyn ≈ 0.10197 kp ≈ 0.22481 Template:Lbf ≈ 7.2330 pdl
1 dyn = 10−5 N ≡ 1 g·cm/s2 ≈ 1.0197 × 10−6 kp ≈ 2.2481 × 10−6 Template:Lbf ≈ 7.2330 × 10−5 pdl
1 kp = 9.80665 N = 980665 dyn gn·(1 kg) ≈ 2.2046 Template:Lbf ≈ 70.932 pdl
1 Template:Lbf ≈ 4.448222 N ≈ 444822 dyn ≈ 0.45359 kp gn·(1 lb) ≈ 32.174 pdl
1 pdl ≈ 0.138255 N ≈ 13825 dyn ≈ 0.014098 kp ≈ 0.031081 Template:Lbf ≡ 1 lb·ft/s2
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units.

## Force measurement

{{#invoke:Portal|portal}}

## Notes

1. Newton's Principia Mathematica actually used a finite difference version of this equation based upon impulse. See Impulse.
2. "It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass." [Emphasis as in the original] Template:Harv
3. "Any single force is only one aspect of a mutual interaction between two bodies." Template:Harv
4. For a complete library on quantum mechanics see Quantum mechanics – References

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