$A$ block is at rest on an inclined plane making an angle $\alpha$ with the horizontal. As the angle $\alpha$ of the incline is increased,the block starts slipping when the angle of inclination becomes $\theta$. The coefficient of static friction between the block and the surface of the inclined plane is:

$A$ block slides down an inclined plane of slope angle $\theta$ with a constant velocity. It is then projected up the plane with an initial velocity $u$. The distance up to which it will rise before coming to rest is

$A$ 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

The time taken by an object to slide down a $45^{\circ}$ rough inclined plane is $n$ times the time it takes to slide down a perfectly smooth $45^{\circ}$ inclined plane. The coefficient of kinetic friction between the object and the inclined plane is:

Two blocks $A$ and $B$ of equal mass are initially in contact when released from rest on an inclined plane. The coefficients of friction between the inclined plane and blocks $A$ and $B$ are $\mu_1$ and $\mu_2$ respectively.

$A$ bob of mass $m$ is attached to a string of length $\ell$ whose other end is tied to a light vertical rod as shown in the figure. The bob is swinging in a horizontal plane with a constant angular speed $\omega$. The vertical rod is supported on a block of mass $M$ which is placed on a rough surface. What is the minimum friction coefficient between the ground and the block for which the block does not slip?