In physics, a force is an external cause responsible for any change of a physical system. For instance, a person holding a dog by a rope is experiencing the force applied by the rope on his hand, and the cause for its pulling forward is the force exercised by the rope. The kinetic expression of this change is, according to Newton's second law, acceleration, non kinetic expressions such as deformation can also occur. The SI unit for force is the newton.

## Elementary Concepts

Force in its most primitive definition can be thought of as that which when acting alone causes an object to accelerate. In a practical sense forces can be divided into two groups: contact forces and field forces. Contact forces require the physical contact of one object with other such as a hammer striking a nail or the force exerted by a gas under pressure—gas produced by exploding gunpowder forces a heavy ball out of a cannon. Field forces on the other hand need no physical medium of contact. Gravity and magnetism are examples of such forces. It should be noted however, that fundamentally all forces are in fact field forces. The force of hammer striking the nail in the previous example turns out to be a clash of the electric forces in both hammer and nail. Nevertheless it is appropriate in some cases to maintain these two classifications for ease of understanding.

## Quantitative definition

In a simple point-like system, one in which objects have no dimensions and cannot rotate or deform, the only change the system can experience is a change of its movement (more precisely of its center of mass); that is, its speed, or more generally its momentum. Since the rise of the atomic theory, any physical system has been considered in classical physics as composed of point-like systems called atoms or molecules. Therefore, all forces can be defined by their effect; that is, by the change of movement they induce on point-like systems. This change of movement can be quantified by the acceleration (the derivative of velocity), but a given force will induce different effects on different point-like systems depending on the system. The discovery by Isaac Newton that a given force will induce an acceleration in inverse proportion to a quantity called the mass of inertia or inertial mass which is independent of the speed of the system is Newton's second law. This law allows us to predict the effect of a force on any point-like system whose mass is known. It is usually written as:

F = dp/dt = d(m·v)/dt = m·a (in the case where m does not depend on t)

where

F is the force (a vector quantity),
p is the momentum,
t is the time,
v is the velocity,
m is the mass, and
a=d²x/dt² is the acceleration, the second derivative with respect to t of the position vector x.

If the mass m is measured in kilograms and the acceleration a is measured in metres per second squared, then the unit of force is kilogram x metre/second squared. This unit is called the newton: 1 N = 1 kg x 1 m/s².

This equation is a system of three second-order differential equations with respect to the three-dimensional position vector which is an unknown function of time. This equation can be solved if F is a known function of x and some of its derivatives and if the mass m is known. Morevover the boundary conditions are required; for example, the values of the position vector and x and the velocity v at the starting time, say t=0.

Of course, this formula is only useful if one knows the numerical values of F and m. The definition above is an implicit definition, arrived at as follows. One defines a reference system (one litre of water) and a reference force (the gravitational force applied by the Earth on it at the altitude of Paris). One takes Newton's second law for granted (one postulates that it is true) and measures the acceleration induced by the reference force on the reference system. This gives us a mass unit (1 kg) and a force unit (the older unit of 1 kilogram-force = 9.81 N). Once this is done, one can measure any force by the acceleration it induces on the reference system and measure the inertial mass of any system by measuring the acceleration induced on this system by the reference force.

Force is often considered a fundamental quantity in physics, but there are more fundamental quantities, such as momentum (p = mass m x velocity v). Energy, measured in joules, is still less fundamental than force and momentum, because it is defined as work, and work is defined in terms of force. The two most fundamental theories of nature - quantum electrodynamics and general relativity - do not contain the concept of force at all.

Although not the most fundamental quantity in physics, force is an important basic mathematical concept from which other concepts, such work and pressure (measured in pascals), are derived. Force is sometimes confused with stress.

## Types of force

There are four known fundamental forces in nature.

• Nuclear forces acting between subatomic particals
• Electromagnetic forces between electric charges
• Weak forces arising from radioactive decay
• Gravitational forces between masses

Quantum field theory accurately models the first three fundamental forces, but does not model quantum gravity. Quantum gravity on a large scale can, however, be described by general relativity.

The four fundamental forces describe every observable phenomenon including the many other forces observed such as: Coulomb's force (the force between electrical charges), gravitational force (force between masses), magnetic force, frictional forces, centripetal, centrifugal, impact force, and spring force, to name a few.

Forces can also be classified into conservative forces and nonconservative forces. Conservative forces are equivalent to the gradient of a potential, and include gravity, electromagnetic force, and spring force. Nonconservative forces include friction and drag.

## Properties of force

Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction.

Forces can be added together using the parallelogram of force. 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 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, with the result that the object remains at rest or moves with a constant velocity.

As well as being added, forces can be 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.

## Forces in theory

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: where

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 to 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.

There are other ways to look at the above definition of force. First, the mass of a body multiplied by its velocity is called its momentum, p, so the above definition is equivalent to: 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. If we graph 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: Many forces are associated with a potential energy field. 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 equal and opposite to the force produced at every point: The derivative of force with respect to time is called yank. Higher order derivatives are sometimes used, but they lack names because of their rarity.

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.

## Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s−2.

### Non-SI units of force and mass

The F=m·a relationship can be used with any consistent units (SI or CGS). If these units are not consistent, a more general form, F=k·m·a, can be used, where the constant k is a conversion factor dependent upon the units being used.

For example, in imperial engineering units, F is measured in "pounds force" or "lbf", m in "pounds mass" or "lb", and a in feet per second squared. In this particular system, one needs 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 standard gee (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 gee 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.

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

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 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.

## Forces in everyday life

Forces are part of everyday life, with examples such as:

• gravity: objects fall, even after being thrown upwards, or slide and roll down
• friction: floors and objects are not extremely slippery
• spring force, objects resist tensile stress, compressive stress and/or shear stress, objects bounce back.
• electromagnetic force: attraction of magnets
• movement created by force: the movement of objects when force is applied.

### Instruments to measure forces

• spring balance
• pivot balance
• forcemeter