Total energy in Simple Harmonic Motion

Energy in Simple Harmonic Motion

The total energy that a body possesses while executing simple harmonic motion is the energy in simple harmonic motion. For example take a pendulum. When it is at its equilibrium position, it is at rest. When it moves towards its extreme position, it is in motion and as soon as it reaches its extreme position, it comes to rest again. Therefore, in order to calculate the energy in simple harmonic motion, we need to calculate the kinetic and potential energy that the body possesses while executing simple harmonic motion.

Kinetic Energy (K.E.) in S.H.M

Kinetic energy is the energy possessed by a body when it is in motion. Consider a body with mass m performing simple harmonic motion along a path AB. Let O be its mean position. Therefore, OA = OB = a ,              

The velocity of the body performing S.H.M. at a distance x from the mean position is given by ,

v= ω √a² – x²

∴ v²   = ω² (a² – x²)

∴ Kinetic energy= 1/2 mv²   = 1/2 m ω² (a² – x²)

As, k/m = ω² 

∴ k = m ω² 

Kinetic energy= 1/2 k (a²– x²) = 1/2ka2 - 1/2kx2 

Potential Energy(P.E.) of Particle Performing S.H.M.

Potential energy is the energy possessed by the body when it is at rest.  Consider a body of mass m performing simple harmonic motion at a distance x from its mean position. As we know that the restoring force acting on the body is F= -kx where k is the force constant.

Now, the body is given infinitesimal displacement dx against the restoring force F. Let the work done to displace the body be dw. Therefore, The work done dw during the displacement is

dw = – fdx = – (- kx)dx = kxdx

Therefore, the total work done to displace the body now from 0 to x is

∫dw=  ∫kxdx = k ∫x dx

Hence Total work done = 1/2 k x2 

The total work done here is stored in the form of potential energy.

Therefore, Potential energy = 1/2 kx2

Total Energy in Simple Harmonic Motion (T.E.)

The total energy in simple harmonic motion is the sum of its potential energy and kinetic energy.

Thus, T.E. = K.E. + P.E.  = 1/2  k ( a2  – x2) + 1/2 kx2 = 1/2 k a2

Hence, T.E.= E = 1/2 ka2

This is the equation of total energy in a simple harmonic motion of a body performing the simple harmonic motion. As k , a2 are constants, the total energy in the simple harmonic motion of a body performing SHM remains constant that means , it is independent of displacement x.

Thus, the total energy in the simple harmonic motion of a particle is:

• Directly proportional to the square of the amplitude of oscillation.

The law of conservation of energy states that energy can neither be created nor destroyed. Therefore,  the total energy in simple harmonic motion will always be constant. However, kinetic energy and potential energy are interchangeable.
Given below is the graph of kinetic and potential energy vs instantaneous displacement.

In the graph, we can see that,

• At the mean position, the total energy in simple harmonic motion is purely kinetic and at the extreme position, the total energy in simple harmonic motion is purely potential energy.
• At other positions, kinetic and potential energies are inter-changeable and their sum is equal to  1/2 k a2.

Also read: what is shm ?