$A$ boy of mass $4 \, kg$ is standing on a piece of wood having mass $5 \, kg$. If the coefficient of friction between the wood and the floor is $0.5$,the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is ...... $N$. (Round off to the Nearest Integer) [Take $g = 10 \, m s^{-2}$]

$A$ block of mass $25 \ kg$ is pulled along a horizontal surface by a force at an angle $45^{\circ}$ with the horizontal. The friction coefficient between the block and the surface is $0.25$. The block travels at a uniform velocity. The work done by the applied force during a displacement of $5 \ m$ of the block is (in $J$)

Three forces of magnitude $F_1, F_2$ and $F_3$ act on a body located at the origin as shown in the figure. The condition that gives zero net force is

Two blocks of mass $2 \ kg$ and $1 \ kg$ are connected by an ideal spring on a rough surface. The spring is unstretched. The spring constant is $8 \ N/m$. The coefficient of friction is $\mu = 0.8$. Now,the $2 \ kg$ block is imparted a velocity $u$ towards the $1 \ kg$ block. Find the maximum value of velocity $u$ of the $2 \ kg$ block such that the $1 \ kg$ block never moves.

$A$ $1.0 \ kg$ block of wood sits on top of an identical block of wood,which sits on top of a flat level table made of plastic. The coefficient of static friction between the wood surfaces is $\mu_1$,and the coefficient of static friction between the wood and plastic is $\mu_2$. $A$ horizontal force $F$ is applied to the top block only,and this force is increased until the top block starts to move. The bottom block will move with the top block if and only if

$A$ block of mass $15 \;kg$ is placed on a long trolley. The coefficient of static friction between the block and the trolley is $0.18$. The trolley accelerates from rest with $0.5 \;m s^{-2}$ for $20 \;s$ and then moves with uniform velocity. Discuss the motion of the block as viewed by
$(a)$ a stationary observer on the ground,
$(b)$ an observer moving with the trolley.