An inextensible cord of negligible mass passes over the rim of a solid disc of mass $M$ and radius $R$. The disc is free to rotate about an axis passing through the centre perpendicular to the plane of the screen, as shown in the figure. Two blocks of masses $M$ and $\widetilde{M} / 2$ are attached to the two free ends of the cord. Assume that there is no slipping of the cord on the disc. The acceleration due to gravity is $g$. What is the value of the angular acceleration of the disc?

Consider the motion of a particle along the $x$-axis. The position of the particle varies with time as $x(t)=\sin ^2(\omega t) \cos ^3(\omega t)$, where $\omega$ is a constant. What is the time period of the motion?

A solid bob of a material having density twice that of water is suspended with a massless and inextensible string of length $L$. The whole set-up is placed inside a water-filled tank. The bob is imparted a horizontal velocity $V_0$ at the lowest point A , while the other end of the string is fixed, such that the bob completes a semi-circular trajectory in the vertical plane. The string becomes slack only when the bob reaches the topmost point C. Assume that the effects of viscosity and water currents are negligible. The acceleration due to gravity is $g$. What is the expression for $V_0$ ?

Consider a solid sphere of radius $R$ floating in a pond with half of the sphere submerged. The sphere is pushed vertically downwards at the topmost point and released, such that it executes a simple harmonic motion. Acceleration due to gravity is $g$. What is the time period of oscillation?