$A$ block of mass $0.18 \ kg$ is attached to a spring of force-constant $2 \ N/m$. The coefficient of friction between the block and the floor is $0.1$. Initially,the block is at rest and the spring is un-stretched. An impulse is given to the block as shown in the figure. The block slides a distance of $0.06 \ m$ and comes to rest for the first time. The initial velocity of the block in $m/s$ is $v = N/10$. Then $N$ is:

$A$ uniform wooden stick of mass $1.6 \,kg$ and length $l$ rests in an inclined manner on a smooth, vertical wall of height $h$ < $l$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $30^{\circ}$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. Find the ratio $h/l$ and the frictional force $f$ at the bottom of the stick. $(g=10 \,m \,s^{-2})$

$A$ block of mass $M = 10\,kg$ rests on a horizontal table. The coefficient of friction between the block and the table is $\mu = 0.05.$ When hit by a bullet of mass $m = 50\,g$ moving with speed $v,$ which gets embedded in it,the block moves and comes to a stop after moving a distance of $2\,m$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H,$ then neglecting energy losses and taking $g = 10\,m/s^2,$ the value of $H$ is close to ................. $km$.

$A$ balloon of mass $M$ is descending at a constant acceleration $\alpha$. When a mass $m$ is released from the balloon,it starts rising with the same acceleration $\alpha$. Assuming that its volume does not change,what is the value of $m$?

$A$ heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m$ $(m \ll M)$. When the car is at rest,the speed of transverse waves in the string is $60 \, ms^{-1}$. When the car has acceleration $a$,the wave-speed increases to $60.5 \, ms^{-1}$. The value of $a$,in terms of gravitational acceleration $g$,is closest to:

$A$ block of mass $m$ moves on a horizontal rough surface with initial velocity $v$. The height of the centre of mass of the block is $h$ from the surface. Consider a point $A$ on the surface.