A solid cylinder of mass $m$ is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:

[The coefficient of static friction, $\mu_{ s },$ is $\left.0.4\right]$

$ABC$ 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

A non uniform cylinder of mass $m$ , length $l$ and radius $r$ is having its cetnre of mass at a distance $l/4$ from the centre and lying on the axis of the cylinder. The cylinder is kept in a liquid of uniform density $\rho $ . The moment of inertia of the rod about the centre of mass is $I$ . The angular acceleration of point $A$ relative to point $B$ just after the rod is released from the position shown in figure is

A disc of radius $20\, cm$ and mass half $kg$ is rolling on an inclined plane. Find out friction force so that disc performs pure rolling.

A uniform meter scale balances at the $40\,cm$ mark when weights of $10\,g$ and $20\,g$ are suspended from the $10\,cm$ and $20\,cm$ marks. The weight of the metre scale is ...... $g$

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 $\alpha>1.5$