If the pulleys shown in the diagram are smooth and massless,and $a_1$ and $a_2$ are the accelerations of the blocks of mass $4 \, kg$ and $8 \, kg$ respectively,then:

Find the acceleration of $B$.

In the arrangement shown in the figure,the block of mass $m=2\,kg$ lies on the wedge of mass $M=8\,kg$. Find the initial acceleration of the wedge if the surfaces are smooth and the pulley and strings are massless.

Two particles $A$ and $B$ are connected by a rigid rod $AB$. The rod slides along perpendicular rails as shown in the figure. The velocity of $A$ to the left is $10\; m/s$. What is the velocity of $B$ (in $m/s$) when the angle $\alpha = 60^{\circ}$?

If the acceleration of block $A$ is $2\,m/s^2$ to the left and the acceleration of block $B$ is $1\,m/s^2$ to the left, then find the acceleration of block $C$.

In the arrangement shown in the figure,$a_{1}, a_{2}, a_{3}$ and $a_{4}$ are the accelerations of masses $m_{1}, m_{2}, m_{3}$ and $m_{4}$ respectively. Which of the following relations is true for this arrangement?