$A$ particle performs rotational motion with an angular momentum $L$. If the frequency of rotation is doubled and its kinetic energy becomes one-fourth,the new angular momentum becomes:

Fill in the blanks:
$(1)$ In rotational motion,the role played by ............ is analogous to the role played by mass in linear motion.
$(2)$ For a rigid body in rotational motion,if a particle at a distance of $10 \ cm$ from the fixed axis of rotation has an angular velocity of $10 \ rad/s$,then the linear velocity of a particle at a distance of $5 \ cm$ from the axis of rotation is ............
$(3)$ The $SI$ unit $J \cdot s^{-2}$ is the unit of the physical quantity ............
$(4)$ The condition for a body to roll without slipping down an inclined plane with friction is ............

$A$ body with a moment of inertia of $3\ kg\cdot m^2$ rotating with an angular velocity of $2\ rad/s$ has the same kinetic energy as a mass of $12\ kg$ moving with a velocity of .......... $m/s$.

One of two identical cylinders,cylinder $A$,rotates at an angular speed of $50 \text{ revolutions per second}$. This rotating cylinder is brought into contact with a second stationary cylinder,$B$. Due to kinetic friction between the two cylinders,the stationary cylinder starts rotating with an angular acceleration,while cylinder $A$ undergoes angular deceleration. If the magnitude of the angular acceleration for both cylinders is $1 \text{ revolution per second}^2$,after how many seconds $(t)$ will the angular speeds of both cylinders become equal?

$A$ uniform rod $AB$ of mass $2 \ kg$ and length $30 \ cm$ is at rest on a smooth horizontal surface. An impulse of force $0.2 \ Ns$ is applied to end $B$. The time taken by the rod to turn through a right angle will be $\frac{\pi}{X} \ s$,where $X = \text{ . . . . . . }$.

$A$ uniform rod of length $2L$ has one end on a horizontal floor. It is inclined at an angle $\alpha$ to the horizontal floor. It falls without slipping,rotating about the point of contact. Its angular velocity when it hits the horizontal floor will be: