$A$ mass of $4\; kg$ rests on a horizontal plane. The plane is gradually inclined until at an angle $\theta = 15^{\circ}$ with the horizontal,the mass just begins to slide. What is the coefficient of static friction between the block and the surface?

The tension $T$ in the string shown in the figure is ..................... $N$.

An ice cube is kept on an inclined plane of angle $30^{\circ}$. The coefficient of kinetic friction between the block and the inclined plane is $(1 / \sqrt{3})$. What is the acceleration of the block? $...... \ m / s ^2$

$A$ 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?

$A$ small block starts sliding down an inclined plane forming an angle $45^{\circ}$ with the horizontal. The coefficient of friction $\mu$ varies with distance $s$ as $\mu = C s^2$,where $C$ is a constant of appropriate dimensions. The distance covered by the block before it stops is:

Two vertical walls are separated by a distance of $2 \ m$. Wall $A$ is smooth while wall $B$ is rough with a coefficient of friction $\mu = 0.5$. $A$ uniform rod is placed between them as shown. The length of the longest rod that can be placed between the walls in equilibrium is equal to