Work - kinetic energy theorem is a generalized description of motion. Kinetic energy is not only related to work with respect to the action of bringing the particle to rest, but to work, in general, that results in change of the speed of particle. We shall, here, formally write work - kinetic energy theorem for a general description of motion, involving change in kinetic energy of the particle resulting from application of a constant external force. At the end of this module, we shall extend the concept to variable force as well.

Let us first do this for a constant force system with resultant force, "F", as applied on a particle or particle like body. The acceleration of the particle is :

A force moves the block on a horizontal surface |
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Let the initial and final velocities of the particle be vi and vf. Then using equation of motion,

Multiplying each term by 1/2 m, we have :

This is the equation, which is known as work - kinetic energy theorem. In words, change in kinetic energy resulting from external force is equal to work done by the force. Equivalently, work done by the force in displacing a particle is equal to change in the kinetic energy of the particle. The above work - kinetic energy equation can be rearranged as :

In this form, work - kinetic energy theorem states that kinetic energy changes by the amount of work done on the particle. We know that work can be either positive or negative. Hence, positive work results increase in the kinetic energy and negative work results decrease in the kinetic energy by the amount of work done on the particle.

### Example 1

Problem : A block of 2 kg is brought up slowly along an incline of length 10 m and height 5 m by applying an external force. At the end of incline, the block is released to slide down to the bottom. If the coefficient of kinetic friction between surfaces is 0.1, find (i) work done by the gravity during round trip (ii) work done by the friction during round trip (iii) work done by the external force and (iii) kinetic energy of the particle at the end of round trip. (consider, g = 10

Solution : This question is structured to bring out finer points about the work done and kinetic energy. Here displacement is along the incline. Thus, we need to consider the components of forces which act along incline plane. Further, we decide the sign of work by determining whether the component of force and displacement are in same direction or opposite to each other.

A block on an incline |
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(i) work done by the gravity during round trip :

The component of gravity along incline is mg sinθ, acting downward. Work done by gravity during up journey is :

Work done by gravity during down journey is :

Total work done during round trip is :

(ii) work done by the friction during round trip :

Friction always acts opposite to displacement. Hence, work done by friction in either direction is negative. Now, kinetic friction is :

(iii) work done by external force during round trip :

External force does work only during up motion. The forces on the block constitue a balanced force system as no acceleration is involved. The net external force is equal to the net component force due to gravity and friction on the block.

(iv) Kinetic energy at the end of round trip :

Initial kinetic energy of the block is zero. The kinetic energy is increased by the work done by the forces acting on the block. Hence, final kinetic energy is :

For understanding purpose and to have an insight, we must understand that net force during up motion is zero. Hence, work done is zero during upward motion. Thus, kinetic energy of the block increases by work done during downward motion only.