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# The phase space diagram for simple Momentum harmonic motion is a circle centred at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and ${{E}_{1}}$ ,${{E}_{2}}$ are the total mechanical energies respectively. Then: (A) ${{E}_{1}}=\sqrt{2}{{E}_{2}}$ (B) ${{E}_{1}}=2{{E}_{2}}$ (C) ${{E}_{1}}=4{{E}_{2}}$ (D) ${{E}_{1}}=16{{E}_{2}}$

Last updated date: 22nd Jun 2024
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Hint: The diagram given to us is the phasor diagram for the simple harmonic motion of the oscillator. The radius of the circle gives us the amplitude of oscillations of the oscillator and the angular velocity for the circle gives us the frequency of oscillation of the oscillator. We need to find the mechanical energy of the oscillator in the two cases.

Formula Used:
$\implies$ $K=\dfrac{1}{2}m{{v}^{2}}$
$\implies$ $P=m\times v$
$\implies$ $K=\dfrac{{{P}^{2}}}{2m}$

The mean position of the oscillator is shown by the y-axis in the given figure. Since the momentum of the two cases is marked along the y-axis, we can say that ${{P}_{1}}=2{{P}_{2}}$ where $P$ denotes momentum of the oscillator.
From the equations $K=\dfrac{1}{2}m{{v}^{2}}$ and $P=m\times v$ , we can say that $K=\dfrac{{{P}^{2}}}{2m}$ where $P$ denotes momentum and $K$ denotes kinetic energy of the particle in oscillation
${{E}_{1}}=\dfrac{{{(2{{P}_{1}})}^{2}}}{2m}$ and ${{E}_{2}}=\dfrac{{{P}_{1}}^{2}}{2m}$ where ${{E}_{1}}$ and ${{E}_{2}}$ are the total mechanical energies of the oscillations
$\dfrac{{{E}_{1}}}{{{E}_{2}}}=\dfrac{4}{1}\Rightarrow {{E}_{1}}=4{{E}_{2}}$
Note: We can alternatively solve this question with the help of the relation of kinetic energy in simple harmonic motion and the amplitude of the oscillation, that is $K=\dfrac{1}{2}m{{\omega }^{2}}{{A}^{2}}$ where $K$ is the kinetic energy, $\omega$ is the frequency of oscillation and $A$ is the amplitude of oscillation. If we approach the question using this method, we won’t even need the diagram. We can just use the statement that their amplitudes are in the ratio $2:1$ .