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PRINCIPLE OF WORK AND ENERGY

In general, the principle of work and energy indicates that the work done on an object from position 1 to position 2 can be calculated as the change in kinetic energy from position 1 to 2:
(1) ΣW1 − 2 =  KE(t2) −  KE(t1) =  KE2 −  KE1
The derivation makes the following assumptions:
  • Mass is constant.
  • All calculations of work and kinetic energy are made in the inertial frame.
  • The body or the individual bodies of a system are rigid (no deformations).
The principle of work and energy is derived from the Newton’s second law of motion. For a body undergoing forces acting on its mass center:
(2) ΣNi = 1 Fc, i = mac
where ΣNi = 1 Fi is the sum of a total of N vector forces applied to the mass center of the body, m is the mass of the body, and aC is the acceleration of the mass center of the body. By representing acceleration as the rate of change of velocity with respect to time and computing the dot product with the velocity of the mass center, vc, the equation becomes
(3) ΣNi = 1 Fc, i  =  m(dvc)/(dt)       ΣNi = 1 Fc, i⋅vc  =  m(dvc)/(dt)⋅vc
Rewriting both sides such that velocity is represented as the rate of change of displacement, ds ⁄ dt, on the left side, and rearranging terms on the right side:
(4) ΣNi = 1 Fc, i⋅(dsc)/(dt)  =  (1)/(2)m(d)/(dt)(vc⋅vc)       ΣNi = 1 Fc, i⋅(dsc)/(dt)  =  (1)/(2)m(d)/(dt)(v2c)
where vc is the magnitude of the velocity vector. Integrating both sides from an initial position, 1, to a final position, 2, with respect to time:
(5) t2⌠⌡t1ΣNi = 1 Fc, i⋅(dsc)/(dt)dt  =  (1)/(2)mt2⌠⌡t1(d(v2c))/(dt)dt       s2⌠⌡ s1ΣNi = 1 Fc, i⋅dsc  =  (1)/(2)mv2c2⌠⌡v2c1d(v2c)
The definition of work for a body where the displacement, s, of the center of mass moves from position 1 to position 2 is:
(6) W1 − 2 =  s2⌠⌡ s1ΣNi = 1 Fc, i⋅dsc
Thus, the principle of work and energy applied to a problem with external forces acting on a single body results in the following equation:
(7) s2⌠⌡ s1ΣNi = 1 Fc, i⋅dsc  =  (1)/(2)m(v2c2 − v2c1)        Work  =  Translational ΔKE
The principle of work and energy can also be extended to applied moments, since moments contribute to rotational kinetic energy. In the case of a problem with only external moments acting on a single body, the sum of the moments about the mass center is equivalent to the rate of change of the angular momentum about the mass center Ḣc:
(8) ΣNi = 1 Mc, i = Ḣc
Following a similar approach as described for forces, for moments, the dot product of the angular velocity, ωc, is used and are integrated both sides, the resulting expression is:
(9) t2⌠⌡t1ΣNi = 1 Mc, iωcdt  =  (1)/(2) ωc(t2)⋅ Hc(t2) + (1)/(2) ωc(t1)⋅ Hc(t1)        work  =  rotational ΔKE      
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