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In physics, the force experienced by a body is its rate of change of momentum with time. Forces are vectors (they specify both a magnitude and direction) and cause a body to accelerate to an amount which is in inverse proportion to its mass. Only four fundamental forces of nature are known: strong, electromagnetic, weak and gravitational (in order of decreasing strength).
Forces were first described by Archimedes in the 3rd century BC but only mathematically grounded by Isaac Newton in the 17th century. Following the development of quantum mechanics it is now thought that particles influence each another through fundamental interactions - so making force a redundant concept.
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. But this theory, although based on the everyday experience of how objects move, was first shown to be unsatsifactory by Galileo as a result of his work on gravity. 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 asserted in his second law of motion that all forces, not just gravity, change the momentum of the body. He defined force as the rate of change of momentum. 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 but with the development of quantum field theory and general relativity it was realised that “force” is redundant concept arising from conservation of momentum (4-momentum in GR and momentum of virtual particles in QFT). Thus currently known fundamental forces are not called forces but “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 nucleii together. The electromagnetic force acts between electric charges and the gravitational force acts between masses.
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. The forces in springs modelled by Hooke's law are the result of electromagnetic forces acting as to return the object to its undistorted position. Centrifugal forces are fictitious forces which arise from rotating frames of reference.
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). When electric charges interact, for instance, they exchange virtual photons.
In general relativity, gravitation is not strictly 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 points. This straight line in space-time is 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. Similarly, planets move in ellipses as they are in an inverse square gravitational field. The time derivative of the changing momentum of the body is what we label as "gravitational force".
- 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 fails to accelerate sideways, 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.
 Quantitative definition
Force is defined as the rate of change of momentum with time:
The quantity (where is the mass and is the velocity) is called the momentum. This is the only definition of force known in physics (first proposed by Newton himself). If the mass m is constant in time, then Newton's second law can be derived from this definition:
where is the acceleration.
This is the form Newton's second law is usually taught in introductory physics courses in order to avoid calculus notation.
All known forces of nature are defined via the above Newtonian definition of force. For example, weight (force of gravity) is defined as mass times acceleration of free fall: w = mg; spring balance force is defined as the force equilibrating certain gravitational force (say, the weight of 1 kg mass near Earth surface results in reaction force of spring equivalent to 9.8 N), etc. Calibration of spring balances (of various kinds) using either gravitational force or motion with known acceleration is important starting procedure in measuring many other forces (such as friction forces, reaction forces, electric forces, magnetic force, etc) in various physics labs.
It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation () is incorrect, and the original form of Newton's second law must be used.
Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction, and four-force is a four-vector in relativity. Vectors (and thus forces) are added together by their components. 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. If the two forces are equal in magnitude but opposite in direction, then the resultant is zero. This condition is called static equilibrium, with the result that the object remains at its constant velocity (which could be zero).
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.
In most explanations of mechanics, force is usually defined only implicitly, in terms of the equations that work with it. Some physicists, philosophers and mathematicians, such as Ernst Mach, Clifford Truesdell and Walter Noll, have found this problematic and sought a more explicit definition of force.
 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 the momentum is given by:
- 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 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 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
 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 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, and include gravity, the electromagnetic force, and the spring force. Nonconservative forces include friction and drag. However, for any sufficiently detailed description, all forces are conservative.
 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.
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
 See also
- Angular momentum
- Conservation law
- Moment map
- Noether's theorem
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