Question

Draw a graph to show the variation of P.E., K.E. and total energy of a simple harmonic oscillator with displacement.

Answer 1

The potential energy (PE) of a simple harmonic oscillator is = `1/2 kx^2`

= `1/2 mω^2x^2`  .....(i)

Where, k = force constant = `mω^2`

When PE is plotted against displacement x, we will obtain a parabola.

When x = 0, PE = 0

When x = ± A, PE = maximum = `1/2 mω^2A^2`

KE of a simple harmonic oscillator = `1/2 mv^2`  .....`[∵ v = ωsqrt(A^2 - x^2)]`

= `1/2 m[ωsqrt(A^2 - x^2)]^2`

= `1/2 mω^2(A^2 - x^2)`  ......(ii)

This is also a parabola if plotting KE against displacement x.

i.e., KE = 0 at x = ± A

And KE = `1/2 mω^2A^2` at x = 0

Now, the total energy of the simple harmonic oscillator = PE + KE  .......[Using equations (i) and (ii)]

= `1/2 mω^2x^2 + 1/2 mω^2 (A^2 - x^2)`

= `1/2 mω^2x^2 + 1/2 mω^2A^2 - 1/2 mω^2x^2`

TE = `1/2 mω^2A^2`

Which is constant and does not depend on x.

Plotting under the above guidelines KE, PE and TE versus displacement x-graph as follows