Two blocks of masses $1 \ kg$ and $2 \ kg$ are connected by a light rod and the system is slipping down a rough incline at an angle of $45^{\circ}$ with the horizontal. The coefficient of kinetic friction at both contacts is $0.4$. If the acceleration of the system is $\alpha \sqrt{2} \ m/s^2$, find the value of $\alpha$. (Use $g = 10 \ m/s^2$)

$A$ 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 $a$?

$A$ $2 \ kg$ brick begins to slide over a surface which is inclined at an angle of $45^{\circ}$ with respect to the horizontal axis. The coefficient of static friction between their surfaces is:

The minimum force required to move a body up an inclined plane is three times the minimum force required to prevent it from sliding down the plane. If the coefficient of friction between the body and the inclined plane is $\frac{1}{2 \sqrt{3}}$, then the angle of the inclined plane is (in $^{\circ}$)

The upper half of an inclined plane of inclination $\theta$ is perfectly smooth,while the lower half is rough. $A$ body starting from rest at the top comes back to rest at the bottom. The coefficient of friction $\mu$ for the lower half is given by:

$A$ uniform rope of length $l$ lies on a table. If the coefficient of friction is $\mu$,then the maximum length $l_1$ of the part of this rope which can overhang from the edge of the table without sliding down is