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.

$A$ person is managing to be at rest between two vertical walls by pressing one wall with his hands and feet and the second wall with his back. The coefficient of friction is $0.5$ between his body and the walls. If the force with which the person pushes the walls is $500 \ N$,then the mass of the person is (Take $g = 10 \ m \ s^{-2}$) (in $kg$)

Three forces acting on a body are shown in the figure. To have the resultant force only along the $y-$ direction,the magnitude of the minimum additional force needed is ........... $N$.

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 $g = 10 \text{ m/s}^2$)

Five persons $A, B, C, D$ and $E$ are pulling a cart of mass $100 \, kg$ on a smooth surface and the cart is moving with an acceleration of $3 \, m/s^2$ in the east direction. When person $A$ stops pulling,it moves with an acceleration of $1 \, m/s^2$ in the west direction. When person $B$ stops pulling,it moves with an acceleration of $24 \, m/s^2$ in the north direction. The magnitude of the acceleration of the cart when only $A$ and $B$ pull the cart,keeping their directions the same as the original directions,is ............ $m/s^2$.

$A$ balloon of total mass $M$ and a fixed size starts coming down with an acceleration $f$ $(f < g)$. The fraction of the total mass of the balloon which must be dropped from it so that it starts going up with an acceleration of $f$ (assuming negligible air resistance) is