The displacements of two particles of same mass executing SHM are represented by the equations $x_1=4 \sin…
The displacements of two particles of same mass executing SHM are represented by the equations
$x_1=4 \sin \left(10 t+\frac{\pi}{6}\right) \text { and } x_2=5 \cos (\omega t) .$
The value of $\omega$ for which the energies of both the particles remain same is
16 unit
6 unit
4 unit
8 unit
Solution
The equation of displacement
$x_1=4 \sin \left(10 t+\frac{\pi}{6}\right)$
The energy of this equation
$E_1=\frac{1}{2} m \omega_1^2 a_1^2$
$=\frac{1}{2} m \times 10 \times 10 \times 4 \times 4$
The second equation of displacement
$x_2=5 \cos (\omega t)$
The energy of this equation
$E_2=\frac{1}{2} m \omega_2^2 a_2^2$
$=\frac{1}{2} m \omega_2^2 \times 5 \times 5$
$\because \quad E_1=E_2$
$\therefore \quad \frac{1}{2} m \omega_2^2 \times 5 \times 5$
$=\frac{1}{2} m \times 10 \times 10 \times 4 \times 4$
$\omega_2^2=\frac{10 \times 10 \times 4 \times 4}{5 \times 5}$
$\omega_2=8$ unit