The total energy of simple harmonic oscillator is proportional to :

Concept:

• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
  • Example: Motion of an undamped pendulum, undamped spring-mass system.
• The potential energy (U) of a particle in simple harmonic motion is given by the formula:

\(\Rightarrow {\rm{U}} = \frac{1}{2}{\rm{k}}{{\rm{x}}^2}\)

Where x = Distance from its mean position and k = spring constant.

• Total Energy: Total energy of a particle is the sum of K.E. and P.E.

Total Energy = K.E. + P.E.

\(Total.Energy. = {1\over 2}mω^2(A^2-x^2)+{1\over 2}mω^2x^2\)

\(Total.Energy. = {1\over 2}mω^2A^2\)

• Total mechanical energy (TE) of a particle executing simple harmonic motion is

\(\Rightarrow TE = \frac{1}{2}{\rm{k}}{{\rm{A}}^2}\)

Explanation:

• The total energy of simple harmonic motion is 

\(Total.Energy. = {1\over 2}mω^2A^2\)

The total energy of a simple harmonic oscillator is proportional to the square of Amplitude.

Additional Information

• Velocity in simple harmonic motion: The relation  between velocity and displacement can be given as:

\(\Rightarrow {\rm{V}} = {\rm{\omega }}\sqrt {{A^2} - {y^2}} \)

Where V = velocity, ω = angular velocity, A = amplitude and y = displacement.

• Kinetic energy (KE) of a particle executing simple harmonic motion is

\(\Rightarrow KE = \frac{1}{2}mv^2\)