Neglecting friction every where, find the acceleration of $M$. Assume $m>m\text{'}$.

 

In the figure shown $P_1$ and $P_2$ are massless pulleys. $P_1$ is fixed and $P_2$ can move. Masses of $A, B$ and $C$ are $\frac{9m}{64} , 2m$ and $m$, respectively. All contacts are smooth and the string is massless. $\theta = {\tan }^{–1}\left(\frac{3}{4}\right)$. Find the acceleration of block $C$ in $\mathrm{m} {\mathrm{s}}^{-2}$.

A system is shown in the figure. All contact surfaces are smooth and string is tight and inextensible. Wedge $A$ moves towards right with speed $10 \mathrm{m} {\mathrm{s}}^{-1}$ and velocity of $B$ relative to $A$ is in downward direction along the incline having magnitude $5 \mathrm{m} {\mathrm{s}}^{-1}$. Find the horizontal and vertical components of the velocity of block $C$.

An object of mass $2 \mathrm{kg}$ is placed at rest in a frame $\left(S_1\right)$ moving with velocity $10\hat{\mathrm{i}} + 5\hat{\mathrm{j}} {\mathrm{ms}}^{-1}$ and having acceleration $5\hat{\mathrm{i}} +10\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$. This object is also seen by an observer standing in a frame $\left(S_2\right)$ moving with velocity $5\hat{\mathrm{i}}+10\hat{\mathrm{j}} {\mathrm{ms}}^{-1}$.

(i) Calculate ‘Pseudo force’ acting on object. Which frame is responsible for this force.

(ii) Calculate net force acting on object with respect to $S_2$ frame.

(iii) Calculate net force acting on object with respect to $S_1$ frame.

In the figure shown, the blocks $A \text{and} C$ are pulled down with constant velocities $u$. Find the acceleration of block $B$.

A system is shown in the figure. Assume that cylinder remains in contact with the two wedges. Find the velocity of cylinder.

In the figure shown, a person pulls a light string with a constant speed $u=10 \mathrm{m} {\mathrm{s}}^{-1}$. The other end of the string is tied to a very small block which moves on a smooth horizontal surface. The block is initially situated at a distance from the pulley which is very large in comparison to $h$. Find the angle $\mathrm{\theta }$ when the block leaves the surface. Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.

A box having mass $400 \mathrm{kg}$ is to slowly slide through $10 \mathrm{m}$ on a horizontal straight track having friction coefficient $0.2$with the box.

(a) Find the work done by the person pulling the box with a rope at an angle $\mathrm{\theta }$ with the horizontal.

(b) Find the work when the person has chosen a value of $\mathrm{\theta }$ which ensures him the minimum magnitude of the force. $[g=10 \mathrm{m} {\mathrm{s}}^{-2}]$

The pulley (uniform disc) shown in figure has, radius $10 \mathrm{cm}$ and moment of inertia about its axis $I=0.5 \mathrm{kg} {\mathrm{m}}^2$ ($B$ and $A$ both move)

(a) Assuming all the plane surfaces are smooth and there is no slipping between pulley and string, calculate the acceleration of the mass $4 \mathrm{kg}$.

(b) The friction coefficient between the block $A$ and the plane below is $\mu =0.5$ and the plane below the $B$ block is frictionless. Assuming no slipping between pulley and string find acceleration of $4 \mathrm{kg}$ block. Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.