The Work-Energy Theorem is an important concept of physics that helps in understanding how forces affect motion by using the relation of work done by forces to changes in an object's kinetic energy.
According to the Work-Energy Theorem, the net force that acts on an object does work on the object equal to the change in its kinetic energy. Mathematically, it can be stated as:
Here:
Work (W) is defined as the dot product of force (F) and displacement(d). Where force is the net force applied to an object and d is the net distance covered by the object due to applied force. Mathematically, it can be expressed as:
Here is the angle between the force and the displacement vector. This formula is used for the constant force.
For Variable force, work done can be calculated as:
Here, ri and rf are the initial and final positions of the object, respectively.
The Kinetic Energy (K) of an object of mass “m” is the energy possessed by any object due to its motion and is moving with a velocity “v”. It can be calculated by the formula:
The change in kinetic energy is:
Where vf = final velocity and vi = initial velocity of the object.
Imagine an object in rectilinear motion under the influence of constant acceleration a, with initial velocity being vi and final velocity vf and it travelled a distance “d”.
To Prove:
Derivation:
Using the third equation of motion,
Here,
u = initial speed of an object
v = final speed of an object
s = distance traversed by that object
Equating the values in the above equation,
…….(1)
Multiply equation 1 by m/2 on both sides,
……..(2)
From Newton’s Second law of motion, we know that force F = ma
Hence, equation 2 will become,
We know that, and Work done W = Fs
Hence,
The Work-Energy theorem can be used to solve many problems related to different concepts of physics as well as real-life problems.
Problem 1: A particle of mass m = 2 kg is moving along the xxx-axis under the influence of a force F(x) = 4x, where x is the position of the particle in meters. The particle starts at rest at x=0 and moves to x = 5 m. Using the Work-Energy Theorem, calculate the final velocity of the particle at x = 5 m.
Solution: Using the formula,
Using the Work-Energy Theorem:
Here, Ki = 0 as the initial velocity is 0.
The kinetic energy of the particle:
Problem 2: A 5 kg block is pushed along a horizontal surface with a constant force of 30 N. The coefficient of kinetic friction between the block and the surface is μk=0.2, and the block moves a distance of 10 meters. Find the final velocity of the block if it starts from rest.
Solution: the frictional force is given by:
Here, N = mg is the normal force. So, the frictional force:
The net force will be
Work done by the net force will be
Using the work-energy theorem:
Problem 3: A 3 kg object is lifted vertically from the ground to a height of 5 meters. Determine the work done on the object and its velocity if it is released from rest and falls under gravity.
Solution: Work done by gravity
Using the Work-Energy theorem:
(Session 2025 - 26)