Question

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies $?_1$ and $?_2$ 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)

• A. $E_1?_1 = E_2?_2$
• B. $\frac{\omega_{2}}{\omega_{1}} = n^{2}$
• C. $?_1?_2 = n^2$
• D. $\frac{E_{1}}{\omega_{1}} = \frac{E^{2}}{\omega_{2}}$

Answer 1

Answer: (D)

$\frac{E_{1}}{\omega_{1}} = \frac{E^{2}}{\omega_{2}}$

Answer 2

$I^{st}$ harmonic oscillator. $\frac{P^{2}}{b^{2}}+\frac{x^{2}}{a^{2}} = 1$ $P^{2} = b^{2} \left(1-\frac{x^{2}}{a^{2}}\right)$ $P = \frac{b}{a} \sqrt{a^{2}-x^{2}}$ $\Rightarrow v = \frac{P}{m}$ $v = \frac{b}{am} \sqrt{a^{2}-x^{2}}$ Comparing $\quad v = \omega\sqrt{A^{2}-x^{2}}$ $\omega_{1} = \frac{b}{am}, A_{1} = a$ $\& \,E_{1} = \frac{1}{2} m \omega^{2}_{1} A^{2}_{1}$ $II^{nd}$ harmonic oscillation $P^{2} + x^{2} = R^{2}$ $P = \sqrt{R^{2}-x^{2}}$ $v = \frac{P}{m}$ $v = \frac{1}{m} \sqrt{R^{2}-x^{2}}\quad\quad\left(v = \omega\sqrt{A^{2}-x^{2}}\right)$ Comparing$\quad\omega_{2} = \frac{1}{m}, A_{2} = R_{1}$ $E_{2} = \frac{1}{2} m \,\omega^{2}_{2} A^{2}_{2}$ $\left(1\right)\quad \frac{\omega_{2}}{\omega_{1}} = \frac{\frac{1}{m}}{\frac{b}{am}}$ $\Rightarrow\quad \frac{\omega_{2}}{\omega_{1}} = \frac{a}{b}\quad\quad$ (given $\frac{a}{b} = n^{2}$) $\frac{\omega_{2}}{\omega_{1}} = n^{2}$ $\left(2\right)\quad \frac{E_{1}}{\omega_{1}} = \frac{1}{2} m\omega_{1} A^{2}_{1}$ $\Rightarrow\quad \frac{E_{1}}{\omega_{1}} = \frac{1}{2} m \left(\frac{b}{am}\right) \left(a\right)^{2}$ $\frac{E_{1}}{\omega_{1}} = \frac{1}{2} ab$ $\&\quad \frac{E_{2}}{\omega_{2}} = \frac{1}{2} m \left(\omega_{2}\right) A^{2}_{2}$ $\frac{E_{2}}{\omega^{2}} = \frac{1}{2} m \left(\frac{1}{m}\right) \left(R\right)^{2}$ $\frac{E_{2}}{\omega_{2}} = \frac{R^{2}}{2}$ the value of ab $= \frac{a^{2}}{n^{2}} \left(\frac{a}{b} = n^{2}\right)$ $\& \quad R^{2} = \frac{a}{n^{2}} \left(\frac{a}{R} = n\right)$ So$\quad \frac{E_{1}}{\omega_{1}} = \frac{E_{2}}{\omega_{2}}$