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? $...... \ m / s ^2$

$A$ piece of ice slides down a rough inclined plane at $\theta=45^{\circ}$ inclination in twice the time that it takes to slide down an identical but frictionless inclined plane. What is the coefficient of friction between ice and incline?

$A$ block of mass $10 \, kg$ is released on a rough inclined plane. The block starts descending with an acceleration of $2 \, m/s^2$. The kinetic friction force acting on the block is ..... $N$ (take $g = 10 \, m/s^2$).

If an inclined plane is made slowly horizontal by reducing its inclination with the horizontal,the component of weight parallel to the plane of a block resting on the inclined plane:

$A$ rectangular box lies on a rough inclined surface. The coefficient of friction between the surface and the box is $\mu$. Let the mass of the box be $m$.
$(a)$ At what angle of inclination $\theta$ of the plane to the horizontal will the box just start to slide down the plane?
$(b)$ What is the force acting on the box down the plane,if the angle of inclination of the plane is increased to $\alpha > \theta$?
$(c)$ What is the force needed to be applied upwards along the plane to make the box either remain stationary or just move up with uniform speed?
$(d)$ What is the force needed to be applied upwards along the plane to make the box move up the plane with acceleration $a$?

As shown in the figure,a body of mass $m$ moving vertically downward with a speed of $3\, m/s$ hits a smooth fixed inclined plane and rebounds with a velocity $v_f$ in the horizontal direction. If the angle of the inclined plane is $30^{\circ}$,the velocity $v_f$ will be: