$A$ block of mass $1 \,kg$ slides down on a rough inclined plane of inclination $60^\circ$ starting from its top. If the coefficient of kinetic friction is $0.5$ and the length of the plane is $1 \,m$,then the work done against friction is ........ $J$.

$A$ block of mass $15\, kg$ is resting on a rough inclined plane as shown in the figure. The block is tied by a horizontal string which has a tension of $50\, N$. The coefficient of friction between the surfaces of contact is $(g = 10\, m/s^2)$.

$A$ 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

$A$ block of mass $10\, kg$ is kept on a rough inclined plane as shown in the figure. $A$ force of $3\, N$ is applied on the block directed down the plane. The coefficient of static friction between the plane and the block is $0.6$. What should be the minimum value of force $P$ applied up the plane,such that the block does not move downward (in $, N$)? (Take $g = 10\, ms^{-2}$)

$A$ body takes time $t$ to reach the bottom of an inclined plane of angle $\theta$ with the horizontal. If the plane is made rough,the time taken now is $2t$. The coefficient of friction of the rough surface is

The minimum force required to start pushing a body up a rough (frictional coefficient $\mu$) inclined plane is $F_{1}$,while the minimum force needed to prevent it from sliding down is $F_{2}$. If the inclined plane makes an angle $\theta$ with the horizontal such that $\tan \theta = 2\mu$,then the ratio $\frac{F_{1}}{F_{2}}$ is: