A small ball of mass $m$ tied to a string is whirled around in a horizontal circle. The length of the string is gradually decreased by pulling it as shown. If $r$ is the radius of the circular path at any instant, the tension will be proportional to

A thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. Another disc of same dimensions but of mass $\frac{1}{4}M$ is placed gently on the first disc coaxially. The angular velocity of the system is

A circular ring $\left(2m,R\right)$ with a small insect of mass $m$ at its periphery, is placed upon smooth horizontal surface as shown in figure. The insect starts moving with velocity $v$ w.r.t. ground along the periphery. The angular velocity of rotation of ring, will be

A uniform road of mass $M$ and length $L$ is hinged at the centre. A particle of mass $m$ and speed $u$ sticks after hitting the end of the rod. Find the angular velocity of the rod just after the collision.

A sphere is rolling without slipping with velocity $V_0$ on a thin plank moving with $3V_0$ as show in figure. Find distance of instantaneous axis of rotation from centre of sphere.

A disc in rolling without slipping on smooth horizontal surface such that velocity of a COM is $V_0$. Find velocity of point shown in the figure using instantaneous axis of rotation.

A rod of mass $m$ and length $l$ is placed on a frictionless horizontal surface. Velocities of its ends are as shown in figure, then total kinetic energy of the rod is

A rod (mass $m$, length $l$) is placed vertically on smooth horizontal surface. Rod is released and after some time velocity of COM of rod became $V_0$ and at this moment rod makes $60°$ with horizontal. Find angular momentum of rod about ICOR.

A uniform solid hemisphere of radius $r$ is joined to a uniform solid right circular cone of base radius $r$. Both have same density. The centre of mass of the composite solid lies on the common face. The height $\left(h\right)$ of the cone is