State whether the following statements are true or false:
$(a)$ $A$ necessary condition for equilibrium is that the acceleration must be zero.
$(b)$ Centripetal force is always in the opposite direction of centrifugal force.
$(c)$ $A$ man is at rest on a perfectly smooth ice surface in the middle of a pond. He can reach the shore by using Newton's first law.
$(d)$ An object is in equilibrium only if it is at rest.

$A$ $300\ kg$ 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 :- ................. $J$

$A$ block of mass $5\, kg$ is $(i)$ pushed in case $(A)$ and $(ii)$ pulled in case $(B)$,by a force $F = 20\, N$,making an angle of $30^o$ with the horizontal,as shown in the figures. The coefficient of friction between the block and floor is $\mu = 0.2$. The difference between the accelerations of the block,in case $(B)$ and case $(A)$ will be ........ $ms^{-2}$ .$(g = 10\, ms^{-2})$

Two blocks ($P$ and $Q$) with masses $2 \text{ kg}$ and $1.5 \text{ kg}$ respectively are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube $(S)$, as shown in the figure. Block $P$ is positioned on the top surface which has no friction, and block $Q$ is in contact with the side surface, which has a coefficient of friction $\mu$. The cube $(S)$ moves towards the right with an acceleration of $g/2$, where $g$ is the gravitational acceleration. During this movement, the blocks $P$ and $Q$ remain stationary relative to the cube. The value of $\mu$ is . . . . . . . (Take $g = 10 \text{ m/s}^2$)

$A$ person with a machine gun can fire $50 \ g$ bullets with a velocity of $240 \ m/s$. $A$ $60 \ kg$ tiger moves towards him with a velocity of $12 \ m/s$. In order to stop the tiger in its track,the number of bullets the person must fire towards the tiger is:

$A$ 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 ................. $km$.