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In physics, a force acting on a body is that which causes the body to accelerate; that is, to change its velocity. The concept appeared first in Newton's second law of motion of classical mechanics, and is usually expressed by the equation:

F = m · a


F is the force, measured in newtons,
m is the mass, measured in kilograms, and
a is the acceleration, measured in metres per second squared.

The field of engineering mechanics relies to a large extent on the concept of force. Scientists have developed a more accurate concept, defining force as the derivative of momentum. Force is not a fundamental quantity in physics, despite the tendency to introduce students to physics via this concept. More fundamental are momentum, energy and stress. Force is rarely measured directly and is often confused with related concepts such as tension and stress.


Forces in applications

Types of forces

Engineers uses many types of force: Coulomb's force between 2 electrical charges, gravity between 2 masses, magnetic force, friction, spring force, ...

However, scientists consider there to be only 4 fundamental forces of nature, with which every observed phenomenon can be explained: the strong nuclear force, the electromagnetic force, the weak nuclear force, and the gravitational force. The former three forces have been accurately modeled using quantum field theory, but a successful theory of quantum gravity has not been developed, although it is described accurately on large scales by general relativity.

Some forces are conservative, while others are not. A conservative force can be written as the gradient of a potential. Examples of conservative forces are gravity, electromagnetic forces, and spring forces. Examples of nonconservative forces include friction and drag.

Properties of force

Forces have an intensity and direction.

Forces can be added together using parallelogram of force. When two forces act on an object, the resulting force (called the resultant) is the vector sum of the original forces. The magnitude of the resultant varies from zero to the sum of the magnitudes of the two forces, depending on the angle between their lines of action. If the two forces are equal but opposite, the resultant is zero. This condition is called static equilibrium, and the object moves at a constant speed (possibly, but not necessarily zero).

While forces can be added together, they can also be resolved into components. For example, an horizontal force acting in the direction of northeast can be split into two forces along the north and east directions respectively. The sum of these component forces is equal to the original force.

Units of measure

The SI unit used to measure force is the newton (symbol N), which is equivalent to kgms−2

Imperial units of force and mass

The relationship F=m·a mentioned above may also be used with non-metric units. If those units do not form a consistent set of units, the more general form F=k·m·a must be used, where the constant k is a conversion factor dependent upon the units used.

For example, in imperial engineering units, F is in "pounds force" or "lbf", m is in "pounds mass" or "lb", and a is in feet per second squared. However, in this particular system, you need to use the more general form above, usually written F=m·a/gc with the constant normally used for this purpose gc = 32.174 lb·ft/(lbf·s2) equal to the reciprocal of the k above.

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

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

When the acceleration of free fall is equal to that used to define pounds force (now usually 9.80665 m/s²), the magnitude of the mass in pounds equals the magnitude of the force due to gravity 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 exert a force of 0.9991 lbf.

The equivalence 1 lb = 0.453 592 37 kg is always true, anywhere in the universe. If you borrow the acceleration 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 which we still see 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.

In conclusion, we have the following conversions:

  • 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

Forces in everyday life

Forces are part of everyday life:

Forces in industry

<to be completed>

Forces in the laboratory

Founding experiments

Instruments to measure forces

<to be completed>

Forces in theory

Force, usually represented with the symbol F, is a vector quantity.

Newton's second law of motion can be formulated as follows:

F = m · a


F is the force, measured in newtons

m is the mass, measured in kilograms

a is the acceleration, measured in metre per second squared

The total (Newtonian) force, in newtons, on an object at any given time is defined as the rate of change of the object's velocity multiplied by the object's mass:

\mathbf{F} = \lim_{T \rightarrow 0 } \frac{m\mathbf{v} - m\mathbf{v}_0}{T}


m is the inertial mass of the particle (measured in kilograms)

vo is its initial velocity (measured in metres per second)

v is its final velocity (measured in metres per second)

T is the time from the initial state to the final state (measured in seconds);

Lim T→0 is the limit as T tends towards zero.

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 experienced 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, and the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion.

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:

\textbf{F}={\Delta \textbf{p} \over \Delta t}

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:

\textbf{F}={d\textbf{p}\over dt}

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:

\textbf{F}=-\nabla U

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.

Non-SI usage of force and mass units

The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM made kilogram-force well defined, by adopting a standard acceleration of gravity for this purpose, 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 vestiges of its use can still occur in:

  • 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 unit of force)
  • Engine torque output (kgf·m expressed in various word order, spelling, 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 disintinguish 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.


Force was first described by Archimedes.

See also

External link

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