# Energy in Simple Harmonic Motion

## How energy change occurs in Simple Harmonic Motion?

The total energy in Simple Harmonic Motion is consisting of two parts –

### Kinetic energy in Simple Harmonic Motion

Velocity of a particle in simple harmonic motion is given by –

v = - \omega A \sin ( \omega t + \phi _ { 0 } )

Hence, kinetic energy of the particle at any position will be –

K = \left ( \frac { 1 }{ 2 } \right ) m v ^ 2

= \left ( \frac {1}{2} \right ) m [ - \omega A \sin ( \omega t + \phi _ { 0 } ) ]^2

= \left ( \frac { 1 }{ 2 } \right ) m \omega ^ 2 A ^ 2 \sin ^ 2 ( \omega t + \phi _ { 0 } )

But, \quad A ^ 2 \sin ^ 2 ( \omega t + \phi _ { 0 } ) = ( A ^ 2 - x ^ 2 )

Therefore, from equation (1), we will get –

K = \left ( \frac {1}{2} \right ) m \omega^2 ( A ^ 2 - x ^ 2 )

For restoring force or force constant in simple harmonic motion –

\omega ^2 = \left ( \frac { k }{ m } \right )

Or, \quad K = \left ( \frac {1}{2} \right ) k ( A ^ 2 - x ^ 2 )

Therefore, kinetic energy of particle in simple harmonic motion has following characteristics –

1. Variation of kinetic energy is periodic in nature. But it is not a simple harmonic activity.
2. Kinetic energy varies from minimum value ( at extreme positions ) to maximum value ( at mean position ) and vice versa. It happens twice in one complete cycle.
3. Therefore, kinetic energy completes 2 cycles in duration of 1 cycle of simple harmonic motion.

### Potential energy in Simple Harmonic Motion

For a particle in simple harmonic motion, the restoring force is given by –

F = - k x

Let, the particle is further displaced through a small distance ( dx ) . Then work done against the restoring force will be –

d W = - F d x = (+ k x ) d x

Therefore, total work done in moving the particle from mean position to a point of displacement ( x )  is given by –

W = \int d W = \int\limits _ { 0 }^{ x } (+ k x) d x

Or, \quad W = k \left [ \frac { x ^ 2 }{ 2 } \right ] _ { 0 }^{ x } = \left ( \frac { 1 }{ 2 } \right ) k x ^ 2

This work is stored as potential energy in the particle.

Therefore, \quad U = \left ( \frac { 1 }{ 2 } \right ) k x ^ 2

= \left ( \frac { 1 }{ 2 } \right ) m \omega ^ 2 x ^ 2

= \frac { 1 }{ 2 } m \omega ^ 2 A ^ 2 \cos ^ 2 ( \omega t + \phi _ { 0 } )

Therefore, for a particle in simple harmonic motion, potential energy has following characteristics –

1. Variation of potential energy is periodic in nature but it is not a simple harmonic activity.
2. Potential energy varies from minimum value ( at mean position ) to maximum value ( at extreme positions ) and vice versa. It happens twice in one cycle of simple harmonic motion.
3. Therefore, potential energy completes 2 cycles in duration of 1 cycle of simple harmonic motion.

### Total energy in Simple Harmonic Motion

At any position of displacement, the total energy of simple harmonic motion is given by –

E = ( K + U )

= \left [ \left ( \frac { 1 }{ 2 } \right ) k ( A ^ 2 - x ^ 2 ) + \left ( \frac { 1 }{ 2 } \right ) k x ^ 2 \right ]

= \left ( \frac {1}{2} \right ) k A^2

= \left ( \frac { 1 }{ 2 } \right ) m \omega ^ 2 A ^ 2

= \left ( \frac {1}{2} \right ) m \left ( \frac {2 \pi}{T} \right )^2 A^2

= 2 \pi ^ 2 m \nu ^ 2 A ^ 2

Because, \quad \left [ \left ( \frac {1}{T} \right ) = \nu \right ]

Therefore, in simple harmonic motion

1. Total mechanical energy is directly proportional to the mass ( m )  of the particle.
2. It is proportional to the square of frequency ( \nu ) .
3. Total mechanical energy is directly proportional to the square of amplitude.
4. It is independent of time ( t ) and displacement ( x ) .

## Variation in total energy along a cycle

Following table shows the variation of kinetic energy ( K ) and potential energy ( U ) at different displacement positions ( x ) .

 Displacement ( x ) ( x = - A ) ( x = 0 ) ( x = + A ) Kinetic energy ( K ) 0 \left ( \frac { 1 }{ 2 } \right ) k A ^ 2 0 Potential energy ( U ) \left ( \frac { 1 }{ 2 } \right ) k A ^ 2 0 \left ( \frac { 1 }{ 2 } \right ) k A ^ 2 Total energy ( E ) \left ( \frac { 1 }{ 2 } \right ) k A ^ 2 \left ( \frac { 1 }{ 2 } \right ) k A ^ 2 \left ( \frac { 1 }{ 2 } \right ) k A ^ 2

The variations of ( K ) and ( U ) with respect to displacement ( x ) is plotted as shown in figure.

From figure, it is evident that –

1. Graph for kinetic energy ( K ) and potential energy ( U ) with respect to the displacement position ( x ) is parabolic curve. But total energy curve is a straight line parallel to the displacement axis.
2. Both kinetic energy ( K ) and potential energy ( U ) are periodic functions of time. The period of each being \left ( \frac {T}{2} \right )
3. At ( x = 0 ) , the potential energy is zero and total energy is the kinetic energy.
4. At ( x = \pm A ) , the kinetic energy is zero and total energy is the potential energy.
5. In each cycle of simple harmonic motion, both kinetic energy and potential energy acquire their peak values twice.