Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $\omega_1$ and $\omega_2$ and have total energies $E_1$ and $E_2$, respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\frac{a}{b}= n ^2$ and $\frac{ a }{ R }= n$, then the correct equation$(s)$ is(are) $Image$
$(A)$ $E_1 \omega_1=E_2 \omega_2$ $(B)$ $\frac{\omega_2}{\omega_1}=n^2$ $(C)$ $\omega_1 \omega_2= n ^2$ $(D)$ $\frac{E_1}{\omega_1}=\frac{E_2}{\omega_2}$
IIT 2015, Medium
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For first oscillator
$E_1=\frac{1}{2} m \omega_1^2 a^2 \text {, and } p = mv = m \omega_1 a = b \Rightarrow \frac{ a }{ b }=\frac{1}{ m \omega_1} .$ $............(i)$
For second oscillator
$E_2=\frac{1}{2} m \omega_2^2 R^2 \text {, and } m \omega_2=1$ $............(ii)$
$\left(\frac{a}{b}\right)=\frac{\omega_2}{\omega_1}=n^2$
$\frac{E_1}{\omega_1^2 a^2}=\frac{E_2}{\omega_2^2 R^2} \Rightarrow \frac{E_1}{\omega_1}=\frac{E_2}{\omega_2}$
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