Rigid Body

On a pulley of mass $M$ hangs a rope with two masses $m_1$ and $m_2\left(m_1>m_2\right)$ tied at the ends as shown in the figure. The pulley rotates without any friction, whereas the friction between the rope and the pulley is large enough to prevent any slipping. Which of the following plots best represents the difference between the tensions in the rope on the two sides of the pulley as a function of the mass of the pulley?

A solid cylinder P rolls without slipping from rest down an inclined plane attaining a speed $v_P$ at the bottom. Another smooth solid cylinder $Q$ of same mass and dimensions slides without friction from rest down the inclined plane attaining a speed $v_Q$ at the bottom. The ratio of the speeds $\left(\frac{v_Q}{v_P}\right)$ is -

A solid sphere rolls without slipping, first horizontally and then up to a point $x$ at height $h$ on an inclined plane before rolling down, as shown in the figure below.

The initial horizontal speed of the sphere is

A rigid ball rolls without slipping on a surface shown below.

Which on the following is the most likely representation of the distance travelled by the ball vs time graph?

A rod of length $l$ slide down along the inclined wall as shown in figure. At the instant shown in figure, the speed of end A is $v$, then the speed of B will be

A rigid body in shape of a triangle has $v_A=5 \mathrm{m}/\mathrm{s}$ $\downarrow$, $v_B=10 \mathrm{m}/\mathrm{s}$ $\downarrow$. Find velocity of point C.