A window (of weight $w$) is supported by two strings passing over two smooth pulleys in the frame of the window in which window just fits in, the other ends of the string being attached to weights each equal to half the weight of the window. One thread breaks and the window moves down. Find acceleration of the window if $\mu$ is the coefficient of friction, and $a$ is the height and $b$ the breadth of the window.

A $160 \mathrm{mm}$ diameter pipe of mass $6 \mathrm{kg}$ rests on a $1.5 \mathrm{kg}$ plate. The pipe and plate are initially at rest when a force $P$ of magnitude $25 \mathrm{N}$ is applied for $0.75 \mathrm{s}$. Knowing that ${\mathrm{\mu }}_{\mathrm{s}}=0.25 \mathrm{and} {\mathrm{\mu }}_{\mathrm{k}}=0.20$ between the plate and both the pipe and the floor, determine;

(a) whether the pipe slides with respect to the plate.

(b) the resulting velocities of the pipe and of the plate.

A uniform disc of mass $m$ and radius $R$ is projected horizontally with velocity $v_0$ on a rough horizontal floor, so that it starts off with a purely sliding motion at $t = 0$. After $t_0$seconds, it acquires a purely rolling motion as shown in the figure.

(a) Calculate the velocity of the centre of mass of the disc at $t_0$.

(b) Assuming the coefficient of friction to be $\mu$, calculate $t_0$. Also calculate the work done by the frictional force as a function of time and  the total work done by it over a time $t$ much longer than $t_0$.

Two uniform thin rods $A\mathit{ }\mathrm{and}\mathit{ }B$ of length $0.6 \mathrm{m}$ each and of masses $0.01 \mathrm{kg} \mathrm{and} 0.02 \mathrm{kg}$, respectively, are rigidly joined, end to end. The combination is pivoted at the lighter end $P$ as shown in figure such that it can freely rotate about the point $P$ in a vertical plane. A small object of mass $0.05 \mathrm{kg}$ moving horizontally hits the lower end of the combination and sticks to it. What should be the velocity of the object so that the system could just be raised to the horizontal position?

A bar of mass $m$ is held as shown between $4$ disks, each of mass $M \mathrm{and} \mathrm{radius} r=75 \mathrm{mm}$. Determine the acceleration of the bar immediately after it has been released from rest, knowing that the normal forces exerted on the disks are sufficient to prevent any slipping and assuming that; In (i) case, the discs are attached to the fixed support on wall. In (ii) case, the discs are attached to the bar.

(a) $m=5 \mathrm{kg} \mathrm{and} M=2 \mathrm{kg}$.

(b) the mass of $M$ of the disks is negligible.

(c) the mass of $m$ of the bar is negligible.

Two thin circular discs of mass $2 \mathrm{kg}$ and radius $10 \mathrm{cm}$ each are joined by a rigid massless rod of length $20 \mathrm{cm}$. The axis of the rod is along the perpendicular to the planes of the disk through their centres. This object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of motion of the truck. The friction with the floor of the truck is large enough, so that object can roll on the truck without slipping. Take $x$-axis as the direction of motion of the truck and $z$-axis as the vertically upward direction. If the truck has an acceleration of $9 \mathrm{m} {\mathrm{s}}^{-2}$, calculate

(a) the force of friction on each disc.

(b) the magnitude and direction of frictional torque acting on each disk about the centre of mass $\mathrm{O}$

A rectangular rigid fixed block has a long horizontal edge. A solid homogeneous cylinder of radius $\mathrm{r}$ is placed horizontally at rest with its length parallel to the edge such that the axis of the cylinder and the edge of the block are in the same vertical plane. There is sufficient friction present at the edge so that a very small displacement causes the cylinder to roll off the edge without slipping. Determine : 

(a) The angle ${\mathrm{\theta }}_{\mathrm{c}}$ through which the cylinder rotates before it leaves contact with the edge.

(b) The speed of the centre of mass of the cylinder before leaving contact with the edge.

(c) The ratio of translational to rotational kinetic energies of the cylinder when its centre of mass is in horizontal line with the edge.