Work Energy Theorem
The law of conservation of energy also expresses a first integral of the equation of motion. Consider an object which is subjected to an external force F so that we have,
m (d v)/dt = F
Integrating above equation with respect to displacement d l, we find
The above equation, known as work-energy theorem, represents the law of conservation of energy. The quantity ½ mv2 is defined as the kinetic energy (or energy of motion) of the object. Left hand side gives the change in kinetic energy of the object when it is displaced from position 1 to 2.
The quantity F . d l is defined as the work done by force F in displacing the object by d l. The integral on the right hand side shows the total work done by the force F in moving the object from position 1 to 2. Work done has the dimensions of energy.
Work-Energy theorem states that change in kinetic energy of an object is equal to the change in some other forms of energy expressed in general as the total work done by all possible forces F acting on the object . For example, if F is the force due to elastic due to elastic stretching of a string, if F is the force of friction, work done represents energy dissipated as heat into environment. If F is sum of both the elastic force and friction, the net work done represents the change in elastic energy as well as heat loss.
If the force F acts along the direction of motion d l, the work done is positive and kinetic energy of the system increases; if force acts opposite to motion, work done is negative and kinetic energy decreases. And, if the force acts perpendicular to motion (as in uniform circular motion), work done is zero and there is no change in kinetic energy. Conversely, if there is no change in kinetic energy of an object in a process, the total work done by all the forces is zero.
Note that work-energy theorem says nothing about the rate of work done, i.e. independent of time taken, total work done is equal to kinetic energy change.
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