$A$ small disk is attached to the end of a light inextensible string,which passes through a hole in a frictionless horizontal tabletop. Initially,the disk moves on a circle of radius $R$ with kinetic energy $K_0$. The string is then slowly pulled so that the disk finally rotates on a circle of radius $\frac{R}{\eta}$. What is the work $W$ done in pulling the string?

$A$ thin circular ring of mass $M$ and radius $r$ is rotating with a constant angular velocity $\omega$. Two particles,each of mass $m$,are attached to the opposite ends of a diameter of the ring. The new angular velocity of the ring will be:

The Sun rotates around its centre once in $27$ days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence (in $days$)? Assume the Sun to be a sphere of uniform density.

$A$ horizontal disc of moment of inertia $4.25 \,kg \cdot m^2$ with respect to its axis of symmetry is spinning counter-clockwise at $15 \,rps$ about its axis,as viewed from above. $A$ second disc of moment of inertia $1.80 \,kg \cdot m^2$ with respect to its axis of symmetry is spinning clockwise at $25 \,rps$ as viewed from above about the same axis and is dropped on top of the first disc. The two discs stick together and rotate as one about their axis of symmetry. The new angular velocity of the system as viewed from above is close to

$A$ solid cylinder of mass $2 \ kg$ and radius $0.2 \ m$ is rotating with an angular velocity of $3 \ rad/s$. $A$ particle of mass $0.5 \ kg$ moving with a velocity of $5 \ m/s$ strikes its periphery tangentially and sticks to it. Find the angular velocity of the system after the collision in $rad/s$.

$A$ child is standing at the center of a rotating platform with hands folded. The kinetic energy of the system is $K$. The child now stretches his hands,doubling the moment of inertia. The kinetic energy of the system will now become ........