Sand is to be piled up on a horizontal ground in the form of a regular cone of a fixed base of radius $R$. The coefficient of static friction between the sand layers is $\mu$. The maximum volume of the sand that can be piled up in the form of a cone without slipping on the ground is

$A$ body is moving up an inclined plane of angle $\theta$ with an initial kinetic energy $E$. The coefficient of friction between the plane and the body is $\mu$. The work done against friction before the body comes to rest is

If $250 \,J$ of work is done in sliding a $5 \,kg$ block up an inclined plane of height $4 \,m$,the value of work done against friction is ............. $J$ $\left(g=10 \,ms^{-2}\right)$.

$A$ block of mass $8 \ kg$ is at rest on a rough inclined plane as shown in the figure. The magnitude of the net force exerted by the surface on the block will be $..... \ N$. (Take $g = 10 \ m/s^2$)

When a body slides down from rest along a smooth inclined plane making an angle of $30^{\circ}$ with the horizontal,it takes time $T$. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance,it takes time $\alpha T$,where $\alpha$ is a constant greater than $1$. The coefficient of friction between the body and the rough plane is $\frac{1}{\sqrt{x}}\left(\frac{\alpha^{2}-1}{\alpha^{2}}\right)$ where $x = .....$.

$A$ uniform rope lies on a horizontal table such that a part of it hangs over the edge. The rope begins to slide down when the length of the hanging part is $25\%$ of its entire length. The coefficient of friction between the rope and the table is