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In physics, force is an influence that may cause a body to accelerate. It may be experienced as a lift, a push, or a pull. The actual acceleration of the body is determined by 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 or deformation of the body. Rotational effects and deformation are determined respectively by the torques and stresses that the forces create.
Force is mathematically defined as the rate of change of the momentum of the body. Since momentum is a vector quantity (has both a magnitude and direction), force also is a vector quantity.
Force was first mentioned by Archimedes in the 3rd century BC but only mathematically defined by Isaac Newton in the 17th century. Following the development of quantum mechanics it is now understood that particles influence each another through fundamental interactions, making force a redundant concept. Only four fundamental interactions are known: strong, electromagnetic, weak (unified into one electroweak interaction in 1970s), and gravitational (in order of decreasing strength).
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 unsatisfactory 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 is credited to give first (and the only) mathematical definition of 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.
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 not called forces but “fundamental interactions”.
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 nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. Pauli's 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.
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 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".
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.
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:
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.
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.
The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s<sup>−2</sup>. 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:
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:
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