The displacements of two particles of same ma | vedclass

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(D) The total energy of a particle executing $SHM$ is given by $E = \frac{1}{2} m \omega^2 A^2$,where $m$ is the mass,$\omega$ is the angular frequenc

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$8$

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(D) The total energy of a particle executing $SHM$ is given by $E = \frac{1}{2} m \omega^2 A^2$,where $m$ is the mass,$\omega$ is the angular frequency,and $A$ is the amplitude.For the first particle,$x_1 = 4 \sin \left(10 t + \frac{\pi}{6}ight)$,the angular frequency $\omega_1 = 10 	ext{ rad/s}$ and amplitude $A_1 = 4 	ext{ units}$.Thus,$E_1 = \frac{1}{2} m (10)^2 (4)^2 = \frac{1}{2} m (100)(16) = 800m$.For the second particle,$x_2 = 5 \cos (\omega t)$,the angular frequency is $\omega$ and amplitude $A_2 = 5 	ext{ units}$.Thus,$E_2 = \frac{1}{2} m \omega^2 (5)^2 = \frac{25}{2} m \omega^2$.Given that the energies are equal,$E_1 = E_2$.$800m = \frac{25}{2} m \omega^2$.$1600 = 25 \omega^2$.$\omega^2 = \frac{1600}{25} = 64$.$\omega = 8 	ext{ units}$.

The displacements of two particles of same mass executing $SHM$ are represented by the equations $x_1 = 4 \sin \left(10 t + \frac{\pi}{6}\right)$ and $x_2 = 5 \cos (\omega t)$. The value of $\omega$ for which the energies of both the particles remain same is (in $\text{ unit}$)

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