# Energy in SHM

Physics

## definition

### Average kinetic energy and potential energy of a SHM

Total energy in SHM is given by, where is the amplitude and remains conserved.

Note:
Average kinetic energy can also be found using
Average potential energy can also be found using

## formula

### Use the relation between restoring force and potential energy

Restoring force is given by:
It is often useful to find the equation of SHM.
Example:
A particle of mass gm is placed in a potential field given by . Find the frequency of oscillation in cycle/sec.
Solution:
Potential energy

## definition

### Relation between restoring torque and potential energy

Restoring torque of a SHM can be found by:
It is often useful in finding equation of SHM and helps in solving problems.

## formula

### Potential energy per unit length at a given point in a travellling sound wave

Potential energy per unit length  at a point is defined as:

p is the sound pressure.
v is the particle velocity in the direction of propagation.
c is the speed of sound.

## example

### Kinetic energy per unit length at a given point in a travelling sound wave

Kinetic Energy of a traveling sound wave is defined as:

Example.
Two pulses in a stretched string whose centres are initially 8 cm apart are moving towards each other as shown in figure. The speed of each pulse is 2 cm/s. After 2 seconds, what will be the total energy of the pulses.

Solution:
After two seconds, the two pulses would nullify each other. As the string now becomes straight, there would be no deformation of the string. In such a situation, there would be no potential energy.

## formula

### Total Potential Energy in one wavelength in a travelling sound wave

Total potential energy of a traveling sound wave is given by:

where,
Linear mass density,
Angular frequency,
Amplitude of wave.

## example

### Derive and use kinetic energy per unit length at a given point in a travelling sound wave

Example:
The velocity of a sound wave in and the wave energy density is then find the amount of energy transferred per unit area per second by the wave in a direction normal to the wave propagation.

Solution:
Energy transferred in normal direction to the wave propogation is also known as intensity

hence