What should be the displacement of a simple pendulum whose amplitude is A, at which potential energy is 1/4 th of the total energy ?
\[ \frac{1}{2} k x^{2}=\frac{1}{4}\left( \frac{1}{2}kA^{2} \right) \]
\[ \frac{1}{2}k x^{2}=\frac{1}{4}\left(kA^{2} \right) x= \frac{A}{2} \]
What is the angular frequency of the system shown in the figure?
The system shown consists of two masses \( M \) connected by a spring with a spring constant \( k \). Since the masses are identical, the angular frequency \( \omega \) of the system for oscillations is given by:
\[
\omega = \sqrt{\frac{k}{\text{reduced mass}}}
\]
In this case, the reduced mass \( \mu \) of the system is given by:
\[
\mu = \frac{M \cdot M}{M + M} = \frac{M}{2}
\]
Thus, the angular frequency \( \omega \) is:
\[
\omega = \sqrt{\frac{k}{M/2}} = \sqrt{\frac{2k}{M}}
\]
Answer:
\[
\omega = \sqrt{\frac{2k}{M}}
\]