A merry-go-round, made of a ring-like platform of radius $R$ and mass$M$, is revolving with angular speed$\text{ω}$. A person of mass$M$ is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round of afterwards is

A uniform solid cylinder is given an angular speed $\omega$ and placed on a rough plate of negligible thickness. The horizontal surface below the plate is smooth. Then the angular speed of the cylinder when it starts pure rolling on the plate will be: [Assume sufficient length of plate]

A sphere $S$ rolls without slipping, moving with a constant speed on a plank $P$. The friction between the upper surface of $P$ and the sphere is sufficient to prevent slipping, while the lower surface of $P$ is smooth and rests on the ground. Initially, $P$ is fixed to the ground by a pin $N$. If $N$ is suddenly removed, 

For next 2 question please follow the same

A disc of mass m and radius R is placed over a plank of same mass m. There is sufficient friction between disc and plank to prevent slipping. A force F is applied at the centre of the disc.

Acceleration of the plank is

A rigid body of radius $R$, either hollow or solid, lies on a smooth horizontal surface. The body is pulled by a horizontal force acting tangentially from the highest point. The distance travelled by the body in the time in which it makes one full rotation is the same, that it will make in one full rotation during pure rolling. The rigid body will be

A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac{3v^2}{4g}$ with respect to the initial position. The object is 

A spool of mass $M=3 \mathrm{kg}$ and radius $R=20 \mathrm{cm}$ has an axle of radius $r=10 \mathrm{cm}$ around which a string is wrapped. The moment of inertia about an axis perpendicular to the plane of the spool and passing through the centre is $\frac{M{R}^2}{2}$. If the coefficient of friction between the surface and the spool is $0.4$ then the maximum tension which can be applied to the string for which the spool doesn't slip, is $\left[g=10 \mathrm{m} {\mathrm{s}}^{-2}\right]$