$A$ solid sphere of mass $2 \ kg$ is rolling on a frictionless horizontal surface with velocity $6 \ m \ s^{-1}$. It collides with the free end of an ideal spring whose other end is fixed. The maximum compression produced in the spring will be (Force constant of the spring $= 36 \ N \ m^{-1}$). (in $m$)

Consider a point $P$ at the contact point of a wheel on the ground,which rolls on the ground without slipping. Find the displacement of point $P$ when the wheel completes half of a rotation (given the radius of the wheel is $1 \ m$).

$A$ uniform solid sphere of radius $R$ and radius of gyration $K$ about an axis passing through the centre of mass,is rolling without slipping. Then the fraction of total energy associated with its rotation will be

$A$ uniform solid cylinder of mass $M = 3 \ kg$ and radius $R = 10 \ cm$ is connected about an axis through the centre of the cylinder to a horizontal spring with spring constant $k = 8 \ N/m$. The cylinder is pulled back,stretching the spring $x = 1 \ m$ from equilibrium. When released,the cylinder rolls without slipping. What is the speed of the center of the cylinder when it returns to equilibrium? .................. $m/s$

Which of the following statements is incorrect for a spherical body rolling without slipping on a rough horizontal ground at rest?

$A$ wheel is rolling straight on ground without slipping. If the axis of the wheel has speed $v$,the instantaneous velocity of a point $P$ on the rim,defined by angle $\theta$ relative to the vertical,relative to the ground will be: