Kinetic Energy-Work Theorem

Kinetic Energy-Work Theorem:

Consider a constant force F acting on a particle of mass m which has an initial velocity ν_A. Let the force cause a displacement x from A to B in the direction of the force. The work done is-

From Newton’s second law, we know that the acceleration ‘a’ which the force ‘F’ generates is given by-

Since ‘F’ is constant, ‘a’ is also constant and the final velocity at B is given by-

Where ν_A is the initial velocity and ν_B is the final velocity.

The kinetic energy is a scalar quantity that depends only on the particle’s speed and not on the direction of motion. Using the symbol K_A and K_B for the kinetic energy that a particle has at A and B respectively, we have from

This shows that the work done by a force acting on a particle is equal to the change in the particle’s kinetic energy. This statement is known as the kinetic energy-work theorem.

Variable Force: The relation (vi) has been derived on the assumption that a constant force F acts throughout the entire journey. Now let us consider the case when the particle of mass m is moving from A to B along a curved path as shown below, under a variable force.

In order to compute the work done as the particle moves from location A to location B, let us divide the distance AB into a large number of small intervals of length Δs. If Δs is small enough, the force F can’t vary much over Δs. Hence, if the average force during the first interval is F₁, during the second interval is F₂, and so on, and these forces are inclined at θ₁ with Δs₁, θ₂ with Δs_2, etc., then

Where ΔW₁, ΔW₂ are the works done during various intervals, ν_A is the initial velocity at A, ν₁, ν₂, ….. are the velocities at the end of first, second … intervals, and ν_n = ν_B, being the speed of the particle at B.

If we add up all the individual work expressions, we get the following relation for the total work done by the applied force.

In terms of the language of calculus this means that

Where K_A and K_B are the kinetic energies of the particle at A and B respectively. The above relation states that the work done by the net force in going from point A to point B equals the kinetic energy at B minus the kinetic energy at A i.e., the change in the kinetic energy is equal to the work done by the net force. Using the language of vectors the work-kinetic energy can be written as-

Calculus Method: Let a variable force act on the particle. As the work done dW in a small displacement dx is Fdx, where F is the instantaneous value of force in the interval x and x + dx, the total work done is given by-