$A$ $10 \, kg$ ball attached to the end of a rigid massless rod of length $L = 1 \, m$ rotates at a constant speed in a horizontal circle of radius $r = 0.5 \, m$ and period $T = 1.57 \, s$ as shown in the figure. The force exerted by the rod on the ball is ........ $N$.

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:

Two carts of masses $200 \, kg$ and $300 \, kg$ on horizontal rails are pushed apart. Suppose the coefficient of friction between the carts and the rails is the same. If the $200 \, kg$ cart travels a distance of $36 \, m$ and stops,then the distance travelled by the cart weighing $300 \, kg$ is ........ $m$.

$A$ spring is compressed between two toy carts of masses $m_1$ and $m_2$. When the toy carts are released,the spring exerts an equal and opposite force on each cart for the same time $t$. If the coefficient of friction $\mu$ between the ground and the carts is the same,find the ratio of the displacements of the two toy carts.

$A$ truck starting from rest moves with an acceleration of $5 \ m/s^2$ for $1 \ s$ and then moves with constant velocity. The velocity $v/s$ time graph for the block on the truck with respect to the ground is: (Assume that the block does not fall off the truck and the coefficient of friction $\mu = 0.2$,$g = 10 \ m/s^2$)

$A$ $300\ kg$ crate is dropped vertically onto a conveyor belt that is moving at $1.20\ m/s$. $A$ motor maintains the belt's constant speed. The belt initially slides under the crate,with a coefficient of friction of $0.400$. After a short time,the crate is moving at the speed of the belt. During the period in which the crate is being accelerated,find the work done by the motor which drives the belt :- ................. $J$