The mass of wedge as shown in the figure is $M$ and that of the block is $m.$ Neglecting friction at all the places and mass of the pulley, calculate the acceleration of wedge. Thread is inextensible.

A block of mass $M$ is connected with a particle of mass $m$ by a light inextensible string as shown in the figure. Assuming all contacting surfaces as smooth, find the acceleration of the wedge after releasing the system.

If the string is inextensible, determine the velocity $u$ of each block in terms of $v$ and $\mathrm{\theta }$.

(a) Figure $\left(a\right): u=....$

(b) Figure $\left(b\right): u=....$

Calculate the accelerations of blocks $A$ and $B$ in cases (a), (b) and (c).

Four blocks and two springs are arranged as shown in figure. The system at rest, determine the acceleration of all the loads immediately after the lower thread keeping the system in equilibrium has been cut. Assume that the threads are weightless, the mass of the pulley is negligible small, and there is no friction at the point of suspension.

The small marble is projected with a velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ in a direction $45°$ from the horizontal $y$-direction on the smooth inclined plane. Calculate the magnitude $v$ of its velocity after $2 \mathrm{s}$.