Physics - SHM Question with Solution | TestHub

The total energy of a simple pendulum is given by $E=\frac{1}{2}m{\omega }^2{A}^2$.

For a simple pendulum, the angular frequency is $\omega =\sqrt{\frac{g}{L}}$.

Substituting $\omega$ in the energy equation, we get $E=\frac{1}{2}m\left(\frac{g}{L}\right)A^2$.

Since the mass $m$, acceleration due to gravity $g$, and linear amplitude $A$ are kept constant, the total energy is inversely proportional to the length of the pendulum, $E\propto \frac{1}{L}$.

When the length is made $\frac{1}{4}$ times ($L^{\prime }=\frac{L}{4}$), the new total energy becomes:

$E^{\prime }=\frac{1}{2}mg\frac{A^2}{(L/4)}=4\left(\frac{1}{2}mg\frac{A^2}{L}\right)=4E$.

Thus, the total energy becomes $4$ times.

Answer: 4 times