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 metre stick is balanced on a knife edge at its centre. When two coins, each of mass $5\; g$ are put one on top of the other at the $12.0 \;cm$ mark, the stick is found to be balanced at $45.0\; cm$. What is the mass of the metre stick?

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

$A$ body is in equilibrium under the influence of a number of forces. Each force has a different line of action. The minimum number of forces required is