In a bicycle,the radius of the rear wheel is twice the radius of the front wheel. If $r_F$ and $r_r$ are the radii,and $v_F$ and $v_r$ are the speeds of the topmost points of the wheels respectively,then:

For a rolling body,the velocities of points $P_1$ and $P_2$ are ${\vec v_1}$ and ${\vec v_2}$,respectively. Which of the following is correct?

An object of mass $10 \ kg$ and radius $0.5 \ m$ is rolling without slipping with a velocity of $2 \ m/s$. Its total kinetic energy is $32.8 \ J$. The radius of gyration of the object is .......... $m$.

$A$ small roller of diameter $20 \ cm$ has an axle of diameter $10 \ cm$. It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle. The scale is now pushed slowly on the axle so that it moves without slipping on the axle,and the roller starts rolling without slipping. After the roller has moved $50 \ cm$,the position of the scale will look like (figures are schematic and not drawn to scale)-

$A$ body is rolling without slipping on a horizontal plane. If the rotational kinetic energy of the body is $50 \%$ of its total kinetic energy,then the body is

Explain why friction is necessary to make the disc in the figure roll in the direction indicated.
$(a)$ Give the direction of frictional force at $B$,and the sense of frictional torque,before perfect rolling begins.
$(b)$ What is the force of friction after perfect rolling begins?