A uniform semicircular ring having mass $m$ and radius $r$ is hanging at one of its ends freely as shown in the given figure. The ring is slightly disturbed so that it oscillates in its own plane. The time period of oscillation of the ring is

A system of two identical rods ($L$-shaped) of mass $m$ and length $l$ are resting on a peg $P$ as shown in the figure. It the system is displaced in its plane by a small angle $\theta$, find the period of oscillations

A particle is subjected to two mutually perpendicular simple harmonic motions such that its $\mathrm{x}$ and $\mathrm{y}$ coordinates are given by $\mathrm{x}=2\mathrm{sin\omega t}; \mathrm{y}=2\sin \left(\mathrm{\omega t}+\frac{\mathrm{\pi }}{4}\right)$

The path of the particle will be
 

Two masses $m_1$ and $m_2$ are suspended together by a massless spring of constant $k$. When the masses are in equilibrium, $m_1$ is removed without disturbing the system. The amplitude of oscillations is,