Shown in the diagram is a system of two bodies,a block of mass $m$ and a disc of mass $4m$,held in equilibrium. If the string $3$ is burnt,find the acceleration of the disc. Neglect the masses of the pulleys $P$ and $Q$. The coefficient of friction between the block and the horizontal surface is $0.5$ and the friction between the disc and the string is zero. (Take $g = 10 \ m/s^2$)

Consider two blocks of mass $m_1 = 5 \, kg$ and $m_2 = 10 \, kg$ placed on a horizontal surface. The coefficient of static friction between the blocks and the surface is $\mu = 0.2$. $A$ horizontal force $F$ is applied to the $10 \, kg$ block. What is the maximum force $F$ (in $N$) that can be applied such that neither block moves? (Take $g = 10 \, m/s^2$)

$A$ ball is dropped from a height of $5 \ m$ onto the floor of an elevator. The ball bounces off the floor. At the time of collision,the elevator is moving upwards with a velocity of $1 \ m/s$. What is the velocity of the ball immediately after the collision in $m/s$?

$A$ $1\; kg$ block situated on a rough incline is connected to a spring of spring constant $100\; N m^{-1}$ as shown in the figure. The block is released from rest with the spring in the unstretched position. The block moves $10\; cm$ down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

At the moment $t=0$, a time-dependent force $F=at$ (where $a$ is a constant equal to $1 \,Ns^{-1}$) is applied to a body of mass $1 \,kg$ resting on a smooth horizontal plane as shown in the figure. If the direction of this force makes an angle $45^{\circ}$ with the horizontal, then the velocity of the body at the moment it leaves the plane is (acceleration due to gravity $=10 \,ms^{-2}$)

$A$ metallic block of mass $20 \ kg$ is dragged with a uniform velocity of $0.5 \ ms^{-1}$ on a horizontal table for $2.1 \ s$. The coefficient of kinetic friction between the block and the table is $0.10$. What will be the maximum possible rise in temperature of the metal block if the specific heat of the block is $0.1 \ cal \ g^{-1} \ ^{\circ}C^{-1}$ (in $^{\circ} C$)? Assume $g = 10 \ ms^{-2}$ and a uniform rise in temperature throughout the whole block. [Ignore absorption of heat by the table]