An ideal spring is connected between two blocks of masses $M$ and $m$. This system can move on a smooth horizontal table. The blocks are brought closer to compress the spring and then released. In the subsequent motion,select the true statement$(s)$:
$(a)$ They move in opposite directions with speeds inversely proportional to their masses.
$(b)$ The ratio of their speeds remains constant.
$(c)$ Linear momentum and energy of the system remain conserved.

Block $A$ of mass $m$ and block $B$ of mass $M$ are connected by a massless spring over a pulley on a rough plane with coefficient of friction $\mu$. $A$ force $F$ is applied on block $A$ to the left. Find the minimum value of $M$ to move the block $A$ towards the right.

In the figures $(i), (ii)$ and $(iii)$ shown, the objects $A, B$ and $C$ have the same mass $m$. The string, spring, and pulley are massless. Object $C$ strikes object $B$ with velocity $u$ in each case and sticks to it. Find the ratio of the velocity of $B$ in case $(i)$ to $(ii)$ to $(iii)$.

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.

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

Imagine the situation in which the given arrangement is placed inside a trolley that can move only in the horizontal direction,as shown in the figure. If the trolley is accelerated horizontally along the positive $x$-axis with $a_0$,then choose the correct statement$(s)$.