In a physical balance working on the principle of moments, when $5\, mg$ weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

The moment of inertia of a solid flywheel about its axis is $0.1\,kg-m^2$. A tangential force of $2\,kg\,wt$. is applied round the circumference of the flyweel with the help of a string and mass arrangement as shown in the figure. If the radius of the wheel is $0.1\,m,$ find the angular acceleration of the solid fly wheel (in $rad/sec^2$)

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 $\alpha>1.5$

A horizontal heavy uniform bar of weight $W$ is supported at its ends by two men. At the instant, one of the men lets go off his end of the rod, the other feels the force on his hand changed to

A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless 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