$A$ horizontal force $F$ is applied at the center of a sphere as shown in the figure. The coefficient of friction between the sphere and the ground is $\mu$. What is the maximum value of $F$ if the sphere does not slip?

$A$ string is wound around a hollow cylinder of mass $5\, kg$ and radius $0.5\, m$. If the string is now pulled with a horizontal force of $40\, N$,and the cylinder is rolling without slipping on a horizontal surface (see figure),then the angular acceleration of the cylinder will be ......... $rad/s^2$. (Neglect the mass and thickness of the string)

$A$ circular disc rolls on a horizontal floor without slipping and the centre of the disc moves with a uniform velocity $v$. Which of the following values can the velocity of a point on the rim of the disc have?

$A$ disc of radius $1\,m$ and mass $4\,kg$ rolls on a horizontal plane without slipping in such a way that its centre of mass moves with a speed of $10\,cm/s$. Its rotational kinetic energy is (in $,J$)

$A$ solid sphere is in rolling motion. In rolling motion,a body possesses translational kinetic energy $(K_t)$ as well as rotational kinetic energy $(K_r)$ simultaneously. The ratio $K_t : (K_t + K_r)$ for the sphere is

$A$ thin hollow cylinder,open at both ends,slides without rotating and then rolls without slipping with the same speed. The ratio of kinetic energies in the two cases will be: