Work Energy Theorem - Definition, Applications, Derivation, Practice Problems, FAQs

What is the difference between a stationary ball on the 5th floor and a moving ball on the same floor? The moving ball, by virtue of its motion, has kinetic energy.

Work is said to be done whenever an object undergoes displacement. A ball for instance, when pushed with a certain force, starts with a certain initial velocity, rolls forward, attains a higher velocity if the force is constantly applied. Since there is a change in its position, work is said to be done on the ball by the external force.The work energy theorem tries to establish a relation between work done and kinetic energy.

Table of contents

Definition of Kinetic energy
Kinetic energy of a System of Particles
Work Energy Theorem
Practice Problems of Work Energy Theorem
FAQs of Work Energy Theorem

Definition of Kinetic energy

The energy possessed by a particle undergoing motion by virtue of its motion is called kinetic energy. It is defined as

K = ½ mv²

Kinetic Energy of a System of Particles

kinetic energy of a system of particles

Fig. shows different particles of masses m₁,m₂ ….. m_n possess velocities v_₁,v_₂ and so on.

Let us consider a system of n number of particles, having masses m₁,m₂....m_n.

Let v₁,v₂.....v_n be the velocities of particles.

Let the KE of the system of particles be K_system. Then

The unit of Kinetic energy is Joule. Its symbol is J.

The dimensional formula is [ML²T^-2]

Work Energy Theorem

According to the work energy theorem, the work done by a force on a body is numerically equal to the kinetic energy change it has undergone. Let us consider a body of mass m , moving with velocity u. Upon applying an external force F, which displaces it through a distance d , accelerates it to a velocity v. Then,

Proof (Derivation)

Since work done W = F × d=m × a × d ; a is the acceleration produced by the force F.

Now by third equation of motion,

i.e Work done = Change in KE.

This can also be proved using the calculus approach.

Since small work done, dW=Fdx , where F is the force applied which displaces it through dx.

Then,

Video explanation

Practice Problems of Work Energy Theorem

1) Calculate kinetic energy of a body of mass 10 g moving with a velocity of 2 ms^-1.

2) A bullet of mass 10 g leaves a rifle with an initial velocity of 1000 ms^-1 and strikes the wall at the same level with a velocity of 500 ms^-1. What is the work done to overcome the resistance of air?

(a) 375 J (b) 3750 J (c) 5000 J (d)500 J

Solution) b

From work energy theorem,

Work done= change in Kinetic energy;

So, the work done to overcome the resistance of air will be 3750 J.

3) A force acts on a 30 g particle in such a way that the displacement of the particle as a function of time is given by , x =3t-4t²+t³, where x is in meters and t is in seconds. What is the work done during the first 4seconds?

(a) 5.28 J (b) 4.50 mJ (c) 49 mJ (d) 530 mJ

Solution)a

Displacement x =3t -4t²+t³

4) In the fig. given below, the block of mass 1 kg has a speed of 0.3 ms^-1 after it has been lowered through a distance of 1 m. Find the coefficient of kinetic friction between the block and the table.