Recall:

Work = W = Fs cosq
Kinetic Energy = KE = (1/2)mv2
Gravitational Potential Energy = PEg = mgh
Work done on an object by conservative forces = change in kinetic energy plus change in potential energy.

Using (conservation of) energy often allows a problem to be solved much more easily than by using forces and Newton's laws.

Potential Energy Stored in a Spring

It is found that springs exhibit a force that is proportional to their compression or elongation, F=kx, where x is the distance from the equilibrium (unstretched) position, and k is the spring constant with units of N/m. This is known as Hooke's law. This force is conservative (as long as the spring is not overstretched). The potential energy stored in a spring that is a distance x from equilibrium is PEs=(1/2)kx2.
Example: P5.29
When a spring of unknown spring constant, k, is compressed 0.120m, it can launch a 20.0g projectile to a height of 20.0m above its starting point. Neglect resistive forces. What is k?

Use conservation of energy. Initially, there is only potential energy of the compressed spring: PEi = 1/2 kx2 = k(0.0072). At the top of the motion, all of the energy is gravitational potential energy: PEf = mgh = (0.0200kg)(9.8m/s2)(20.0m) = 3.92J. Since all the forces are conservative, the inital and final energies must be equal, or: (0.0072)k = 3.92J, such that k=544N/m.

Non-conservative Forces and Work-Energy Theorem

If non-conservative forces are present, then the sum of KE and PE is not constant. The difference between the final and initial energies equals the work done by the non-conservative forces.

Wnc = (KEf+PEf)-(KEi+PEi)

Example: P5.32
An 80.0N box is pulled 20.0m up a 30 degree incline by an applied force of 100N parallel to the incline. If the coefficient of kinetic friction between box and incline is 0.220, calculate the change in kinetic energy of the box.

Use Wnet = KEf-KEi, and the fact that Wnc = Fs cosq. We assume that initially the box is not moving, and assign the zero for gravitational potential energy to the bottom of the incline. Wnc=-mkmg cos30 s = -(0.220)(80.0N)(20.0m)cos30 = -305J. Wg = -PEf = -mgh = -mgs sinq = -(80.0N)(20.0m)sin30 = -800J. WF = Fs cosq = (100N)(20.0m) = 2000J. KEf = WF + Wnc + Wg = 2000-305-800 = 895J.

As a general rule, it is found that energy is conserved in the universe. Energy comes in many forms, and can generally be converted from one form to another, but there no evidence of energy non-conservation has yet been found. This observation is now so fundamental, that when a situation arises where it seems that energy is not conserved, other possible explanations are always sought first.