Question

When the displacement in S.H.M is one-third of amplitude what fraction of the total energy is potential and what fraction is kinetic energy?

Answer 1

Hint:We are asked to find out how much part of total energy is potential energy and kinetic energy. First recall the formulas for total energy and potential energy in S.H.M or simple harmonic motion, use those to find out the part of total energy which is potential energy. Then use conservation of energy to find out the part for kinetic energy.

Complete step by step answer:
Given, displacement is one-third of amplitude. Let \[A\] be the amplitude of the S.H.M, then displacement will be
\[x = \dfrac{1}{3}A\] ………(i)
Total energy in S.H.M is given as,
\[T.E = \dfrac{1}{2}m{\omega ^2}{A^2}\] ……………(ii)
Where \[m\] is the mass of the particle executing S.H.M, \[\omega \] is the frequency and \[A\] is the amplitude.
To find out what fraction of total energy is potential energy, let’s find out the potential energy. Potential energy in S.H.M is given as,
\[P.E = \dfrac{1}{2}m{\omega ^2}{x^2}\]
Putting the value of \[x\] in the above equation we get,
\[P.E = \dfrac{1}{2}m{\omega ^2}{\left( {\dfrac{A}{3}} \right)^2} \\
\Rightarrow P.E = \dfrac{1}{2}m{\omega ^2}\dfrac{{{A^2}}}{9} \\
\Rightarrow P.E = \dfrac{1}{9}\left( {\dfrac{1}{2}m{\omega ^2}{A^2}} \right) \]
Now, substituting equation (ii) in the above equation we get,
\[P.E = \dfrac{1}{9}\left( {T.E} \right)\] ……....(iii)
Therefore, we observe that potential is \[\dfrac{1}{9}\] of total energy.
From conversation of energy we get,
\[K.E + P.E = T.E\] …...(iv)
where \[K.E\] is the kinetic energy.
Now substituting the value of potential energy from equation (iii) in equation (iv), we get
\[K.E + \dfrac{1}{9}\left( {T.E} \right) = T.E \\
\Rightarrow K.E = T.E - \dfrac{1}{9}\left( {T.E} \right) \\
\therefore K.E = \dfrac{8}{9}T.E \]

Therefore, kinetic energy is \[\dfrac{8}{9}\] of total energy and potential energy is \[\dfrac{1}{9}\] of total energy.

Note:S.H.M or Simple harmonic motion is a type of oscillatory motion and also periodic. In case of S.H.M the restoring force is directly proportional to the displacement from the mean position. One important point you should remember is that all S.H.M are oscillatory motions but all oscillatory motions may not be S.H.M.