A cylinder rests in a supporting carriage, as shown. The side $AB$ of the carriage makes an angle $30°$ with the horizontal and the side $BC$ is vertical. The carriage lies on a fixed horizontal surface and is being pulled towards left with a horizontal acceleration $a$. The magnitude of the normal reactions exerted by sides $AB$ and $BC$ of the carriage on the cylinder is $N_{\mathrm{AB}}$ and $N_{\mathrm{BC}}$, respectively (neglect friction everywhere). Then, as the magnitude of acceleration $a$ of the carriage is increased, choose the correct statement.

A ball is suspended on a thread from the ceiling of a car. The brakes are applied and the speed of the car changes from $5 \mathrm{m} {\mathrm{s}}^{-1}$ to $\frac{5}{3} \mathrm{m} {\mathrm{s}}^{-1}$, during a time interval of $3 \mathrm{s}$. Find the angle at which the thread with the ball, will deviate from the vertical.

A block of mas $2 \mathrm{kg}$ slides down the face of a smooth $45°$ wedge of mass $9 \mathrm{kg}$ as shown in figure. The wedge is placed on a frictionless horizontal surface. Determine the acceleration of the wedge.

In the figure shown, $A$ and $B$ are free to move, and they are released from rest. All the surfaces are smooth. $\left(0<\theta <90°\right)$

In the pulley system shown in the figure, the block $C$ is going up at $2 \mathrm{m} {\mathrm{s}}^{-1}$ and the block $B$ is going up at $4\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}$. Find the velocity of block $A$.

A system is shown in the figure. A man is pulling the rope from both sides with constant speed $u$. Find the speed of the block.

A rod $AB$ is shown in the figure. The end $A$ of the rod is fixed on the ground. Block is moving with velocity $\sqrt{3} \mathrm{m} {\mathrm{s}}^{-1}$ towards the right. The velocity of the end $B$ of the rod, when the rod makes an angle of $60°$ with the ground is

In the figure shown, find the value of $\text{θ}$ [assume string to be tight].