$A$ mass $2 \sqrt{3} \,kg$ is acted upon by two forces which are inclined to each other at $60^{\circ}$ and each of magnitude $1 \,N$. The acceleration of that mass in $SI$ system is $\left[\sin 30^{\circ}=\cos 60^{\circ}=0.5\right]$ (in $\,m / s^{2}$)

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 $\mu + \sqrt{3}$.

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. When the downward acceleration of the elevator becomes equal to $g$,then

Two masses $A$ and $B$ of $10\,kg$ and $5\,kg$ respectively are connected by a string passing over a frictionless pulley fixed at the corner of a table. The coefficient of friction between mass $A$ and the table is $0.2$. The minimum mass of $C$ that must be placed on $A$ to prevent it from moving is equal to $...........\,kg$.

Two carts are pushed apart by the explosion of a charge placed between them. $A$ cart of mass $100 \ kg$ stops after traveling a distance of $18 \ m$. What distance will a cart of mass $300 \ kg$ travel before stopping? The coefficient of friction $\mu$ between the carts and the ground is the same for both.

The masses of blocks $A$ and $B$ are $m$ and $M$ respectively. Between $A$ and $B$,there is a constant frictional force $F$ and $B$ can slide on a smooth horizontal surface. $A$ is set in motion with velocity $v_0$ while $B$ is at rest. What is the distance moved by $A$ relative to $B$ before they move with the same velocity?