Two ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$. In one experiment, the…
Two ends of a stretched wire of length $L$ are fixed at $x = 0$ and $x = L$. In one experiment, the displacement is $y_1 = A \sin \left( \frac{\pi x}{L} \right) \sin \omega t$ and energy is $E_1$ and in another experiment, its displacement is $y_2 = A \sin \left( \frac{2\pi x}{L} \right) \sin 2\omega t$ and energy is $E_2$, then
$E_2 = E_1$
$E_2 = 2E_1$
$E_2 = 4E_1$
$E_2 = 16E_1$
Solution
$E = \text{Energy density} \times \text{Volume} = \frac{1}{2} \rho \omega^2 A^2 sL$
or $E \propto \omega^2$, rest all quantities are common for both the waves.
$\omega_2 = 2\omega$ and $\omega_1 = \omega$
$\therefore E_2 = 4 E_1$