Question

What fraction of the total energy is kinetic energy when the displacement of a simple oscillator is half of its amplitude.

Answer 1

Hint: Firstly, we have to take total energy. Then, calculate kinetic energy. After that, take the fraction of the amount of total energy to be kinetic energy and find the desired value of the fraction.

Formula used:
Here, we are using two formulas. One is kinetic energy. That is, \[\dfrac{1}{2}m{{\omega }^{2}}\left( {{a}^{2}}-{{x}^{2}} \right)\]and the other is of total energy. That is,\[\dfrac{1}{2}m{{\omega }^{2}}{{a}^{2}}\]
In both of them. ‘m’ is mass,’\[\omega \]’ is angular velocity and ‘a’ is amplitude. Whereas, ‘$x$’ is displacement.

Complete step-by-step solution:
Since, Displacement ($x$) is half of the amplitude of a simple oscillator.
So, $x=\dfrac{a}{2}$
Now, Total energy = \[\dfrac{1}{2}m{{\omega }^{2}}{{a}^{2}}\]
Also, kinetic energy when displacement ($x$) is :-

= \[\dfrac{1}{2}m{{\omega }^{2}}\left( {{a}^{2}}-{{x}^{2}} \right)\]
\[=\dfrac{1}{2}m{{\omega }^{2}}\left[ {{a}^{2}}-{{(\dfrac{a}{2})}^{2}} \right]\]
 \[=\dfrac{3}{4}\left[ \dfrac{1}{2}m{{\omega }^{2}}{{a}^{2}} \right]\]
Therefore, fraction of the total energy to be kinetic energy is :-
\[\Rightarrow \dfrac{Kinetic}{Total}\dfrac{Energy}{Energy}\]
\[\Rightarrow \dfrac{\dfrac{3}{4}\left[ \dfrac{1}{2}m{{\omega }^{2}}{{a}^{2}} \right]}{\dfrac{1}{2}m{{\omega }^{2}}{{a}^{2}}}\]
So, the result is \[\dfrac{3}{4}\]
This means that the \[\dfrac{3}{4}\]is the amount of the total energy to be kinetic energy.When the displacement of a simple oscillator is half of its amplitude.
Additional Information: The total energy in simple harmonic motion is defined as the sum of two energies. That is its potential energy which is defined as the energy possessed by the particle when it is at rest at some height and its kinetic energy which is defined as the energy possessed by an object when it is in oscillatory motion.

Note:Here, we are taking both total energy and kinetic energy equations for Simple Harmonic Motion only, not for general rectilinear motion. Also, put the values in the equation as well as find the calculation properly.