# Energy Equations of a Mass Attached to a Spring in SHM

Physics

## example

### Write kinetic energy as a function of time in SHM

Kinetic energy as a function of time in SHM:

E =

## formula

### Write potential energy as a function of time in SHM

Formula for potential energy as a function of time in SHM is:

## formula

### Total energy as a function of time in SHM

Total energy as a function of time in SHM:

Total energy = (Independent of time)

## example

### Conservation of total mechanical energy to find amplitude

Example: Potential energy of a particle in SHM along axis is given by:  .Here, is in joule and in meter. Total mechanical energy of the particle is Mass of the particle is Find the amplitude of oscillation.

Solution:
minimum potential energy at mean position
At extreme position
Total mechanical energy

Hence and are the extreme positions.
Amplitude of oscillation

## example

### Write kinetic energy, potential energy and total energy of a mass attached to a spring in SHM

Kinetic Energy in SHM:
Potential Energy is :
Total Energy is:

Example: The potential energy of a simple pendulum in its resting position is J and its mean kinetic energy is J. What will be its total energy at any instant?

Solution:
The total energy of the system remains constant. Since it is given that P.E at rest is J, the total energy must be J as K.E at rest is 0. As total energy of the system is conserved in SHM.

## example

### Problem on kinetic energy, potential energy and total energy of a mass attached to a spring in SHM

Example: A mass is attached to a spring of stiffness executing SHM. It has amplitude and velocity at the equilibrium position is . Find the total energy of this spring mass system.

Solution:
At the extreme position of the spring it has only potential energy since velocity is zero:
At the equilibrium position it has no stretch in the spring.
Kinetic energy at this instant:
At any instant of time during the motion:
Total energy = KE + PE = =