Calculate the acceleration (in $m/s^{2}$) of the block and trolley system shown in the figure. The coefficient of kinetic friction between the trolley and the surface is $0.05$. (Given: $g = 10\; m/s^{2}$,mass of the string is negligible and no other friction exists).

In the diagram,$BAC$ is a rigid fixed rough wire and angle $BAC$ is $60^o$. $P$ and $Q$ are two identical rings of mass $m$ connected by a light elastic string of natural length $2a$ and elastic constant $k = \frac{mg}{a}$. If $P$ and $Q$ are in equilibrium when $PA = AQ = 3a$,then the least coefficient of friction between the ring and the wire is $\mu$. Find the value of $\mu + \sqrt{3}$.

Four blocks $A, B, C$ and $D$ of masses $6 \,kg, 3 \,kg, 6 \,kg$ and $1 \,kg$ respectively are connected by light strings passing over frictionless pulleys as shown in the figure. The strings $P$ and $Q$ are horizontal. The coefficient of friction between the horizontal surface and the block $B$ is $0.2$ and the blocks $A$ and $B$ move together. If the system is released from rest, then the tension in the string $Q$ is (Acceleration due to gravity, $g=10 \,ms^{-2}$) (in $\,N$)

In the figure,two blocks $M$ and $m$ are tied together with an inextensible and light string. The mass $M$ is placed on a rough horizontal surface with coefficient of friction $\mu$ and the mass $m$ is hanging vertically against a smooth vertical wall. The pulley is frictionless. Choose the correct statement$(s)$ related to the tension $T$ in the string.

Give the magnitude and direction of the net force acting on a stone of mass $0.1\; kg$,
$(a)$ just after it is dropped from the window of a stationary train,
$(b)$ just after it is dropped from the window of a train running at a constant velocity of $36\; km/h$,
$(c)$ just after it is dropped from the window of a train accelerating with $1\; m/s^2$,
$(d)$ lying on the floor of a train which is accelerating with $1\; m/s^2$,the stone being at rest relative to the train.
Neglect air resistance throughout.

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.