Potential Energy: Suppose a body is moving in a gravitational field, then the work done by the gravity on the body is equal to , where is the change in the potential energy of the body. Similarly if the body is moving against a spring force, the work done by the spring on the body is where is the change in the potential energy of the spring. If is the work done by the forces (other than gravity and spring forces) during the motion of the body from configuration 1 to 2, the following relation will hold good,
where is the increase in kinetic energy. The above equation is called work energy equation. It states that if the forces are conservative, the work done by the forces (other than gravity and spring forces, which have been accounted for in writing potential energy expression) will equal to change in kinetic energy plus change in potential energy.
Virtual Work : The work done by the forces for an arbitrary assumed small and kinematically consistant displacement is called virtual work and is denoted by .Using the work-energy relation,
Now, where Si is the displacement of the particle and ai is the acceleration. Similarly,
Interconnected Bodies: For a system of interconnected rigid bodies, we can say
The force F includes only external forces(internal forces between interconnecting bodies are equal and opposite and must move with the mass center of the system, hence they contribute no work).
An example: TWo bodies of weight W1 and W2 have been conencted by a string. The friction forces of the bodies during motion are indicated by F1 and F2. The bodies are pulled by applying a force P as shown in the figure. If the initial speed of bodies is u, find out the speed after the force P has moved a distance S.
Solution: The work done by the external force (including friction but excluding gravity force).
This work will increase the kinetic energy of the system of bodies.
Increase in kinetic energy =
From this the final velocity v may be found.