An infinite number of masses are placed on a frictionless table and they are connected via massless strings. Their masses follow the sequence $m, \frac{m}{2}, \frac{m}{6}, \ldots, \frac{m}{n!}, \ldots$ and they are further connected to a mass $m$ that hangs over a massless pulley. The acceleration of the hanging mass is

$A$ block of mass $M$ is placed on a horizontal surface and is tied with an inextensible string to a block of mass $m$,as shown in the figure. $A$ block of mass $m_0$ is also placed on $M$. The surface under $M$ is smooth,while the vertical surface under $m$ is rough with a coefficient of friction $\mu$. The minimum value of $\mu$ for which the block $m$ remains stationary is:

$A$ 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 $\,m / s^{2}$)

$A$ metal block is resting on a rough wooden surface. $A$ horizontal force applied to the block is increased uniformly with time,i.e.,$F = kt$. Which of the following curves correctly represents the velocity $v$ of the block as a function of time $t$?

Match the items in Column-$I$ with those in Column-$II$.

Column-$I$	Column-$II$
$(1)$ Newton's third law of motion	$(a)$ $\vec{F}_{12} = -\vec{F}_{21}$
$(2)$ Law of conservation of momentum	$(b)$ $\Delta \vec{p} = 0$
	$(c)$ $\vec{F}_{12} = \vec{F}_{21}$

In the figure,a ladder of mass $m$ is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu_1$ and that between the floor and the ladder is $\mu_2$. The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2$. If the ladder is about to slip,then