$A$ disc and a ring of the same mass are rolling. If their kinetic energies are equal,then the ratio of their velocities will be:

Consider a body of mass $1.0 \ kg$ at rest at the origin at time $t=0$. $A$ force $\overrightarrow{F}=(\alpha t \hat{i}+\beta \hat{j})$ is applied on the body,where $\alpha=1.0 \ Ns^{-1}$ and $\beta=1.0 \ N$. The torque acting on the body about the origin at time $t=1.0 \ s$ is $\vec{\tau}$. Which of the following statements is (are) true?
$(A)$ $|\vec{\tau}|=\frac{1}{3} \ Nm$
$(B)$ The torque $\vec{\tau}$ is in the direction of the unit vector $+\hat{k}$
$(C)$ The velocity of the body at $t=1 \ s$ is $\overrightarrow{v}=\frac{1}{2}(\hat{i}+2 \hat{j}) \ ms^{-1}$
$(D)$ The magnitude of displacement of the body at $t=1 \ s$ is $\frac{1}{6} \ m$

$A$ uniform thin wooden plank $AB$ of length $L$ and mass $M$ is kept on a table with its $B$ end slightly outside the edge of the table. When an impulse $J$ is given to the end $B$,the plank moves up with the centre of mass rising a distance $h$ from the surface of the table. Then,

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. Which graph represents the variation of $I$ with respect to $R$,keeping $M$ constant?

$A$ uniform body of mass $M$ and radius $R$ has a small mass $m$ attached at its edge as shown in the figure. The system is placed on a perfectly rough horizontal surface such that mass $m$ is at the same horizontal level as the centre of the body. It is assumed that there is no slipping at point $A$. If $I_A$ is the moment of inertia of the combined system about the point of contact $A$,then the normal reaction at point $A$ just after the system is released from rest is ........ $N$. ($M = 6 \ kg$,$m = 2 \ kg$,$I_A = 4 \ kg \ m^2$,$R = 1 \ m$,$g = 10 \ m/s^2$)

$A$ ring of radius $R$ rolls without sliding with a constant velocity $u$. The radius of curvature of the path followed by any particle of the ring at the highest point of its path will be