$A$ solid cylinder of mass $3 \ kg$ is rolling on a horizontal surface with velocity $4 \ m/s$. It collides with a horizontal spring whose one end is fixed to a rigid support. The force constant of the spring is $200 \ N/m$. The maximum compression produced in the spring will be (assume the collision between the cylinder and the spring is elastic). (in $m$)

$A$ rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is:

$A$ flywheel is an important part of a steam engine because it

$A$ spherical rigid ball is released from rest and starts rolling down an inclined plane from height $h=7 \, m$,as shown in the figure. It hits a block at rest on the horizontal plane (assume elastic collision). If the mass of both the ball and the block is $m$ and the ball is rolling without sliding,then the speed of the block after collision is close to ............. $m/s$.

State whether the following statements are true or false:
$(1)$ In the law of conservation of momentum, the center of mass velocity remains constant.
$(2)$ "If the resultant of internal forces on a system is zero, then the total linear momentum of the system remains constant." This is the statement of the law of conservation of linear momentum.
$(3)$ The position of the center of mass of a rigid body must always be inside the rigid body.
$(4)$ All particles of a rigid body undergoing rotational motion have the same linear velocity.

$A$ uniform cylinder of radius $1 \,m$, mass $1 \,kg$ spins about its axis with an angular velocity $20 \,rad/s$. At a certain moment, the cylinder is placed into a corner as shown in the figure. The coefficient of friction between the horizontal wall and the cylinder is $\mu$, whereas the vertical wall is frictionless. If the number of rounds made by the cylinder is $5$ before it stops, then the value of $\mu$ is (acceleration due to gravity, $g=10 \,m/s^2$)