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 thin rod of length $4l,$ mass $4m$ is bent at the points as shown in the figure. What is the moment of inertia of the rod about the axis passing point $O$ and perpendicular to the plane of the paper?

A uniform $rodAB$ of mass $m$ and length $l$ at rest on a smooth horizontal surface. An impulse $P$ is applied to the end $B\mathit{.}$ The time taken by the rod to turn through a right angle is :-

An extended object consists of two point masses, ${\mathrm{m}}_1$ and ${\mathrm{m}}_2,$ connected via a rigid massless rod of length $\mathrm{L},$ as shown in the figure $\left(\mathrm{i}\right).$ Initially the object is at rest. The object can rotate about an axis perpendicular to the page through the midpoint of the rod. Two time-varying tangential forces, ${\mathrm{F}}_1$ and ${\mathrm{F}}_2,$ are applied to ${\mathrm{m}}_1$ and ${\mathrm{m}}_2,$ respectively as shown in the figure $\left(\mathrm{ii}\right).$

fig. (i)

fig. (ii)