If a ring, a disc, a solid sphere and a cylinder of same radius rolls down on inclined plane, the first one to reach the bottom will be :-

A cubic block of mass $m$ and side length $b$ is placed on a smooth floor. A smooth and rigid rod of length $L$ and with negligible mass is leaning against the block. A sphere of mass $M$ is attached to the upper end of the rod. The lower end of the rod is fixed at point-$O,$ but the rod can rotate freely around the point within the vertical plane. Initially the angle between the rod and the floor is $\mathrm{\alpha }$ while the system is at rest. After releasing, find the speed of the block when the angle between the rod and the floor is $\mathrm{\beta }$

A disc of mass $M$ and radius $R$ rolls on a rough horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v,$ the height to which the disc will rise will be :-

A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity $v {\mathrm{ms}}^{-1}.$ If it is to climb the inclined surface then $v$ should be:-

A rod hinged at one end is released from the horizontal position as shown. in the figure. When it becomes vertical its lower half separ tes without, exerting any reaction at the breaking point. Then the maximum angle $\text{'}\mathrm{\theta }\text{'}$ made by the hinged upper half with the vertical is

A small uniform tube is bent into a circular tube of radius $R$ and kept in the vertical plane. Equal volumes of two liquids of densities $\rho$ and $\sigma (\rho >\sigma )$ fill half of the tube as shown. $\theta$ is the angle which the radius passing through the interface makes with the vertical. The value of $\theta$ is :-

A non uniform cylinder of mass $m,$ length $L$ and radius $r$ is having its centre of mass at a distance $\frac{L}{4}$ from the centre and lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho ,$ The moment of inertia of the rod about the centre of mass is $I.$ The angular acceleration of point $A$ relative to point $B$ just after the rod is released from the position shown in figure is :