A force in physics is any action that causes an object's velocity to change; that is, to accelerate. A force originates from within a
field, such as an electro-static field (caused by static electrical charges), electro-magnetic field (caused by moving charges), or gravitational field (caused by mass), among others.
Newton was the first to mathematically express the relationship between
force and
momentum. Some physicists interpret
Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's second law": : \mathbf{F} = {\mathrm{d}\mathbf{p} \over \mathrm{d}t} = {\mathrm{d}(m \mathbf{v}) \over \mathrm{d}t}. The quantity
mv is called the (
canonical)
momentum. The net force on a particle is thus equal to the rate of change of the momentum of the particle with time. Since the definition of acceleration is , the second law can be written in the simplified and more familiar form: : \mathbf{F} = m \mathbf{a} \, . So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an
ordinary differential equation, which is called the
equation of motion. As an example, assume that friction is the only force acting on the particle, and that it may be modeled as a function of the velocity of the particle, for example: : \mathbf{F}_{\rm R} = - \lambda \mathbf{v} \, , where
λ is a positive constant, the negative sign states that the force is opposite the sense of the velocity. Then the equation of motion is : - \lambda \mathbf{v} = m \mathbf{a} = m {\mathrm{d}\mathbf{v} \over \mathrm{d}t} \, . This can be
integrated to obtain : \mathbf{v} = \mathbf{v}_0 e^{{-\lambda t}/{m}} where
v0 is the initial velocity. This means that the velocity of this particle
decays exponentially to zero as time progresses. In this case, an equivalent viewpoint is that the kinetic energy of the particle is absorbed by friction (which converts it to heat energy in accordance with the
conservation of energy), and the particle is slowing down. This expression can be further integrated to obtain the position
r of the particle as a function of time.
Newton's third law can be used to deduce the forces acting on a particle when in a closed system. If it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, −F, on A. For
conservative forces, this means that the line integral around a closed loop is zero. The strong form of Newton's third law requires that F and −F act along the line connecting A and B, and these forces are defined as
central forces. However, central forces are an approximation since objects that are at rest are only at rest with respect to one another. This limitation to Newton's third law can be shown using the
Coulomb force, where charges must remain stationary with respect to a nonaccelerating frame of reference. When dealing with non-central forces like the
Lorentz force, the weak form of Newton's third law is used by identifying
conservation of momentum. Illustrations of the weak form of Newton's third law can be found for magnetic forces like the Lorentz force while discussing the curl or cross product of vectors. Thus, the forces acting on objects cannot be identified without accounting for relative acceleration and direction by utilizing reference frames.
Work and energy If a constant force
F applied to a particle displaces it from position
rinitial to
rfinal, then the
work done, W, by the force is defined as the
scalar product of that force and the displacement vector : : W = \mathbf{F} \cdot \Delta \mathbf{r} \, . More generally, if the force varies as a function of position as the particle moves from
r1 to
r2 along a path
C, the work done on the particle is given by the
line integral : W = \int_C \mathbf{F}(\mathbf{r}) \cdot \mathrm{d}\mathbf{r} \, . If the work done in moving the particle from
r1 to
r2 is the same no matter what path is taken, the force is said to be
conservative.
Gravity is a conservative force, as is the force due to an idealized
spring, as given by
Hooke's law. The force due to
friction is non-conservative. The
kinetic energy Ek of a particle of mass
m travelling at speed
v is given by : E_\mathrm{k} = \tfrac{1}{2}mv^2 \, . For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles. The
work–energy theorem states that for a particle of constant mass
m, the total work
W done on the particle as it moves from position
r1 to
r2 is equal to the change in
kinetic energy Ek of the particle: : W = \Delta E_\mathrm{k} = E_\mathrm{k_2} - E_\mathrm{k_1} = \tfrac{1}{2} m \left(v_2^{\, 2} - v_1^{\, 2}\right) . Conservative forces can be expressed as the
gradient of a scalar function, known as the
potential energy and denoted
Ep: : \mathbf{F} = - \mathbf{\nabla} E_\mathrm{p} \, . If all the forces acting on a particle are conservative, and
Ep is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force : \mathbf{F} \cdot \Delta \mathbf{r} = - \mathbf{\nabla} E_\mathrm{p} \cdot \Delta \mathbf{r} = - \Delta E_\mathrm{p} \, . The decrease in the potential energy is equal to the increase in the kinetic energy : -\Delta E_\mathrm{p} = \Delta E_\mathrm{k} \Rightarrow \Delta (E_\mathrm{k} + E_\mathrm{p}) = 0 \, . This result is known as
conservation of energy and states that the total
energy, : \sum E = E_\mathrm{k} + E_\mathrm{p} \, , is constant in time. It is often useful, because many commonly encountered forces are conservative. == Lagrangian mechanics ==