During simple harmonic motion, energy is constantly exchanged between two forms: kinetic and potential.

The potential energy could be in the form of:

• Gravitational potential energy (for a pendulum)
• Elastic potential energy (for a horizontal mass on a spring)
• Or both (for a vertical mass on a spring)

Speed v is at a maximum when displacement x = 0, so:

The kinetic energy is at a maximum when the displacement x = 0 (equilibrium position).

Therefore, the kinetic energy is 0 at maximum displacement x = x0, so:

The potential energy is at a maximum when the displacement (both positive and negative) is at a maximum x = x0 (amplitude).

A simple harmonic system is therefore constantly converting between kinetic and potential energy.

When one increases, the other decreases and vice versa, therefore:

The total energy of a simple harmonic system always remains constant and is equal to the sum of the kinetic and potential energies

The kinetic and potential energy of an oscillator in SHM vary periodically

The key features of the energy-time graph:

• Both the kinetic and potential energies are represented by periodic functions (sine or cosine) which are varying in opposite directions to one another
• When the potential energy is 0, the kinetic energy is at its maximum point and vice versa
• The total energy is represented by a horizontal straight line directly above the curves at the maximum value of both the kinetic and potential energy
• Energy is always positive so there are no negative values on the y axis

Note: kinetic and potential energy go through two complete cycles during one period of oscillation

This is because one complete oscillation reaches the maximum displacement twice (positive and negative)

The key features of the energy-displacement graph:

• Displacement is a vector, so, the graph has both positive and negative x values
• The potential energy is always at a maximum at the amplitude positions x0 and 0 at the equilibrium position (x = 0)
• This is represented by a ‘U’ shaped curve

The kinetic energy is the opposite: it is 0 at the amplitude positions x0 and maximum at the equilibrium position x = 0

• This is represented by an ‘n’ shaped curve
• The total energy is represented by a horizontal straight line above the curves

Average kinetic energy and potential energy of a SHM

The total energy in SHM is given by,

 E=12​mω2A2

 where A is the amplitude and remains conserved.                                         
E=K + U

Kavg=Uavg=E2=14mω2A2

Note:

Average kinetic energy can also be found using 

Kavg=1T0TKdt

Average potential energy can also be found using 

Uavg=1T0TUdt

Use the relation between restoring force and potential energy

Restoring force is given by:

 F=- d Ud X   

It is often useful to find the equation of SHM.

Example:
A particle of mass 10 gm is placed in a potential field given by V=(50x2+100)J/kg. Find the frequency of oscillation in cycle/sec.
Solution:

Potential Energy U=mV

U=10-250x2+100

 F=- d Ud X​=-(100x)10-2  

m2x=-(10010-2) x

1010-32x=-10010-2x

  2=100,   =10
f= 2=102=5

Relation between restoring torque and potential energy

Restoring torque of an SHM can be found by:

=-dUd

It is often useful in finding the equations of SHM and helps in solving problems.

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

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

w=pvc
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

The Kinetic Energy of a travelling sound wave is defined as: