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This article is about the concept of force in physics. For all other uses see Force (disambiguation).

In physics, a net force acting on a body causes that body to accelerate (i.e. to change its velocity vector). The SI unit used to measure force is the newton.

See also engineering mechanics:

Statics Where the sum of the forces acting on a body in static equilibrium (motionless, Acceleration=0) is zero. F=MA=0

Dynamics The sum of the forces acting on a body or system over time is non zero with a resulting set of accelerations defined by detailed analysis of equations derived from F=MA.

Force was first described by Archimedes. The total (Newtonian) force on a point particle at a certain instant in a specified situation is defined as the rate of change of its momentum:

F = Limit as T goes to zero of (mv - mvo)/T

Where m is the inertial mass of the particle, vo is its initial velocity, v is its final velocity, and T is the time from the initial state to the final state.

Force was so defined in order that its reification would explain the effects of superimposing situations: If in one situation, a force is experienced by a particle, and if in another situation another force is experience by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, along with the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion.

Since force is a vector it can be resolved into components. For example, a 2D force acting in the direction North-East can be split in to two forces along the North and East directions respectively. The vector-sum of these component forces is equal to the original force.

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The content of above definition of force can be further explicated. First, the mass of a body times its velocity is designated its momentum (labeled p). So the above definition can be written:

<math>\textbf{F}={\Delta \textbf{p} \over \Delta t}</math>

If F is not constant over Δt, then this is the definition of average force over the time interval. To apply it at an instant we apply an idea from Calculus. Graphing p as a function of time, the average force will be the slope of the line connecting the momentum at two times. Taking the limit as the two times get closer together gives the slope at an instant, which is called the derivative:

<math>\textbf{F}={d\textbf{p}\over dt}</math>

With many forces a potential energy field is associated. 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. The potential field is defined as that field whose gradient is minus the force produced at every point:

<math>\textbf{F}=-\nabla U</math>

While force is the name of the derivative of momentum with respect to time, the derivative of force with respect to time is sometimes called yank. Higher order derivates can be considered, but they lack names, because they are not commonly used.

In most expositions 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.

Relationships between force units and mass units

In the relationship

F = m×a,
which is derived from Newton’s second law of motion, F is the force in newtons, m the mass in kilograms and a the acceleration in meters per second squared. To a physicist, the kilogram is a unit of mass, but in common usage "kilogram" is a shorthand for "the weight of a one kilogram mass at sea level on earth". At sea level on earth, the acceleration due to gravity (a in the above equation) is 9.807 meters per second squared, so the weight of one kilogram is 1 kg × 9.807 m/s² = 9.807 N.

To distinguish these two meanings of "kilogram", the abbreviations "kgm" for kilogram mass (i.e. 1 kg) and "kgf" for kilogram force (i.e. 9.807 N) are sometimes used. These are only informal terms and are not recognized in the SI system of units.

Imperial units of force

The relationship F = m×a mentioned above may also be used with non-metric units.

For example, in imperial engineering units, F is in “pounds force” or "lbf", m is in "pounds mass" or "lbm", and a is in feet per second squared.

As with the kilogram, the pound is colloquially used as both a unit of mass and a unit of force or weight. 1 lbf is the force required to accelerate 1 lbm at 32.174 ft per second squared, since 32.174 ft per second squared is the acceleration due to terrestrial gravity at sea level.

Another imperial unit of mass is the slug, defined as 32.174 lbm. It is the mass that accelerates by one foot per second squared when a force of one lbf is exerted on it.

Conversion between SI and imperial units of force

At sea level on earth, the magnitude of lbm exactly equals the magnitude of lbf, and the magnitude of kgm exactly equals the magnitude of kgf. This equivalency is only true at the surface of the earth, and does not hold up when acceleration other than that of the standard acceleration of gravity (that at the sea level of Earth) is used.

In other words, your mass and force exerted on the ground equal the same number in pounds (that is, lbm and lbf) on Earth at sea level. Since kgf and lbf are units of force, they are invariant, and the equivalence 1 kgf = 2.2046 lbf is always true. However, the conversion 1 kgm = 2.2046 lbm is true only on Earth at sea level.

The concept of weight, unlike force and mass, depends on the environment in which the weighing is done. It can be assumed that this is at sea level on Earth, unless other conditions are stated. Thus one pound mass (lbm) weighs one pound (lbf), and one kilogram mass (kgm) weighs one kilogram force (kgf). Further, an item with a weight of 10 lbf has a mass of 10 lbm and also a mass of 0.3108 slugs (= 10 lbm divided by 31.174 lbm per slug).

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 weight (on Earth at sea level) of 10 kgf has a mass of 10 kgm and also a mass of 1.0197 metric slugs (= 10 kgm divided by 9.807 kgm per metric slug).

An even rarer unit of force called the "imperial newton" is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lbm times one foot per square second, we have (1/32.174 =) 0.0311 lbf = 1 lbm times 1 foot per square second = 1 imperial newton. Thus 1 lbf = 32.174 imperial newtons.

In conclusion, we have the following conversions, with “metric slugs” used very infrequently, and “imperial newtons” virtually never used.

1 kgf = 9.807 newton 1 metric slug = 9.807 kgm

1 lbf = 32.174 imperial newtons 1 slug = 32.174 lbm

1 kgf = 2.2046 lbf

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