$A$ uniform sphere of radius $R$ and mass $m$ is placed on an inclined plane which makes an angle $45^{\circ}$ to the horizontal. For which of the following values of the coefficient of friction does the sphere roll without slipping? Select the incorrect option.

$A$ solid sphere and a disc,having the same mass and radius,roll down from rest from the same height on a smooth inclined plane. The ratio of the time taken by them is:

$A$ hollow sphere of mass $m$ filled with a non-viscous liquid of same mass $m$ is released on a slope inclined at angle $\theta$ with the horizontal. The friction between the sphere and the slope is sufficient to prevent sliding,and frictional forces between the inner surface of the sphere and the liquid are negligible. After descending a certain height,the ratio of translational and rotational kinetic energies is found to be $x:y$. Find the numerical value of the expression $(x+y)_{min}$.

What is the velocity of a sphere starting from rest and rolling without slipping down an inclined plane of vertical height $h$?

$A$ cylinder of mass $M_c$ and a sphere of mass $M_s$ are placed at points $A$ and $B$ of two inclines,respectively (See Figure). If they roll on the incline without slipping such that their accelerations are the same,then the ratio $\frac{\sin \theta_c}{\sin \theta_s}$ is

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: $(i)$ a ring of radius $R$,$(ii)$ a solid cylinder of radius $\frac{R}{2}$,and $(iii)$ a solid sphere of radius $\frac{R}{4}$. If,in each case,the speed of the center of mass at the bottom of the incline is the same,the ratio of the maximum heights they climb is: