A mass $m$ hangs with the help of a string wrapped around a pulley on a firctionless  bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect  uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the  pulley, is:-

A uniform disc with mass $M=4\,kg$ and radius $R=$ $10\,cm$ is mounted on a fixed horizontal axle as shown in figure. A block with mass $m =2\,kg$ hangs from a massless cord that is wrapped around the rim of the disc. During the fall of the block, the cord does not slip and there is no friction at the axle. The tension in the cord is_______ $N$

$\left(\right.$ Take $\left.g =10\,ms ^{-2}\right)$

A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in Figure. The angles made by the strings with the vertical are $36.9^{\circ}$ and $53.1^{\circ}$ respectively. The bar is $2\; m$ long. Calculate the distance $d$ of the centre of gravity of the bar from its left end.

$ABC$ is an equilateral triangle with $O$ as its centre. $\vec F_1, \vec F_2 $and $\vec F_3$ represent three forces acting along the sides $AB, BC$ and $AC$ respectively. If the total torque about $O$ is zero then the magnitude of  $\vec F_3$ is

The left end of a massless stick with length $l$ is placed on the corner of a table, as shown in Fig. A point mass $m$ is attached to the center of the stick, which is initially held horizontal. It is then released. Immediately afterward, what normal force does the table exert on the stick?

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A block of weight $\alpha W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2 \sqrt{2}) W$. Which of the following statement($s$) is(are) correct ?

$(A)$ The vertical component of reaction force at $O$ does not depend on $\alpha$

$(B)$ The horizontal component of reaction force at $O$ is equal to $W$ for $\alpha=0.5$

$(C)$ The tension in the rope is $2 W$ for $\alpha=0.5$

$(D)$ The rope breaks if $\alpha>1.5$