Work and Energy

© Fernando Caracena, 1 November 2015

Note that there is a parallel discussion of this topic, which is restricted to algebraic arguments in the post,  An Algebraic discussion of Work and Energy. The following discussion is best followed by those who know calculus well.


Early in the development of classical physics, physicists developed the concept of energy and of the conserved flow of energy in the universe. These concepts became increasingly appropriate as various new forms of energy were discovered. For example, the energy of the movement of a mass, m, was called kinetic energy, and defined as follows:

KE= ½ m v2;                              (1a)


KE= p2 /(2m) ;                         (1b)

where the momentum of a mass in motion is

p  = m v .                                   (1c)

Note that bold characters represent vectors.

Recall how Newton's laws apply to the force on an object of mass m:

                            F = d(m v)/dt ,                      (2a)

which is the most general form of Newton's equations of motion that applies to systems of variable mass, such as rockets and comets. Alternatively, (2a) is written in terms of momentum, p, as follows:

                           F = dp/dt .                               (2b)

Applied to changes in the kinetic energy of a mass, we can see that the force exerted on that mass (2b) plays a role in changing the mass' s kinetic energy,

dKE/dt =( p/m) dp/dt            (3a)

dKE/dt = v F .                            (3b)

The last equation above (3b) is further simplified by multiplying  both sides by the infinitesimal time interval, dt,

dKE       = dr F ,                         (4a)

which removes time from the equation.  If the force is a function only of position and not time, then (4a) can be integrated to give the change in kinetic energy produced by such a force when the mass travels from position r1 to position r2 ,

KE2  - KE1  = ∫12 dr F .                (4b)

Further, if the integral between position r1 and position ris independent of the path taken, then the force is called conservative, in which case (4b) establishes the kinetic energy of a mass as a function of position. Note that the transfer of energy by a force is called work, in physics

W = ∫12 dr F .                                  (5)

KE2 =  KE1  + W                                  (6)


Even if a force is not conservative, such as friction, the work done by that force represents a transfer of energy from one form to another. In the case of friction, large scale motion is degraded into random motions of atoms and molecules, which constitute heat. And heat is a form of energy. In fact, there are many forms of energy.

So far, physicists cannot find any sources or sinks of energy in the universe. The transfer of energy from one form to another may look like a source of energy, or a sink for energy; but if you look hard enough you can always trace an unbroken chain of energy exchange in a zero sum fashion, the total of which does not change in this universe.


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