$A$ 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 is placed on a rough horizontal plane. $A$ time-dependent horizontal force $F = Kt$ acts on the block,where $K$ is a positive constant. The acceleration-time graph of the block is:

$A$ string of negligible mass going over a clamped pulley of mass $m$ supports a block of mass $M$ as shown in the figure. The force on the pulley by the clamp is given by

Two wedges each of mass $600 \ g$ are placed next to each other on a rough horizontal surface. The coefficient of static friction between the wedges and the surface is $0.4$. $A$ cube of mass $M$ is balanced on the wedges as shown in the figure. If there is no friction between the cube and wedges,the largest mass $M$ of the cube that can be balanced without motion of the wedges is . . . . . . $kg$.

$A$ racing car travels on a track (without banking) $ABCDEFA$. $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R = 100 \, m$. The coefficient of friction on the road is $\mu = 0.1$. The maximum speed of the car on straight paths is $50 \, m/s$. Find the minimum time for completing one round.

Three forces $\vec{F}_1=(2 \hat{i}+4 \hat{j}) \,N$,$\vec{F}_2=(2 \hat{j}-\hat{k}) \,N$,and $\vec{F}_3=(\hat{k}-4 \hat{i}-2 \hat{j}) \,N$ are applied on an object of mass $1 \,kg$ at rest at the origin. The position of the object at $t=2 \,s$ will be: