In physics, force is an 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 the giant phallus 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 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 times acceleration :

(or, more generally, as the rate of change of momentum). This approach is disparaged by the large majority of textbooks.^[1] By making this a definition of force, all empirical content is removed from the law. In fact, the 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 remains valid. But in order to be conserved, momentum must be redefined as:

where

is the velocity and

is the speed of light.

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

where the Lorentz factor

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 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

for use in relativity through the use of four-vectors. This relation is correct in relativity when is the four-force, m is the invariant mass, and 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 is defined as that field whose gradient is equal and opposite to the force produced at every point:

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

Conservative forces

Main article: Conservative force

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 eminating from spherically symmetric potentials. Examples of this follow:

For gravity:

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

For electrostatic forces:

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

For spring forces:

where 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 cvt

See also

Angular momentum Conservation law g-force Impulse

Inertia Moment map Momentum

Noether's theorem Physics Velocity

References