$A$ block of mass $4 \ kg$ is at rest on a rough inclined plane making an angle of $\theta$ with the horizontal. The coefficient of static friction between the block and the plane is $0.5$ and the frictional force on the block is $14.14 \ N$. Find the value of $\theta$. (in $^{\circ}$)

$A$ child weighing $25$ kg slides down a rope hanging from the branch of a tall tree. If the force of friction acting against him is $2$ $N$,the acceleration of the child is ........ $m/s^2$. (Take $g = 9.8$ $m/s^2$)

$A$ block of mass $m$ is stationary on a rough plane of mass $M$ inclined at an angle $\theta$ to the horizontal,while the whole setup is accelerating upwards at an acceleration $a$. If the coefficient of friction between the block and the plane is $\mu$,then the force that the plane exerts on the block is

$A$ disc arranged in a vertical plane has two grooves of the same length directed along the vertical chord $AB$ and the chord $CD$ as shown in the figure. Particles slide down along $AB$ and $CD$ starting from rest. The ratio of the time $t_{AB}/t_{CD}$ 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:

$A$ cubic block of mass $m$ is sliding down on an inclined plane at $60^{\circ}$ with an acceleration of $\frac{g}{2}$. The value of the coefficient of kinetic friction is: