If $\vec F$ is the force acting on a particle having position vector $\vec r$ and $\vec \tau$ be the torque of this force about the origin, then:

A uniform rod of length $200~ \text{cm}$ and mass $500~ \text g$ is balanced on a wedge placed at $40~ \text{cm}$ mark. A mass of $2~\text{kg}$ is suspended from the rod at $20~ \text{cm}$ and another unknown mass $m$ is suspended from the rod at $160~\text{cm}$ mark as shown in the figure. What would be the value of $m$ such that the rod is in equilibrium?
(Take $g=10~( \text {m/s}^2)$

A non-uniform bar of weight $W$ is suspended at rest by two strings of negligible weight as shown in the 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. The distance $d$ of the center of gravity of the bar from its left end is:
(Take sin$36.9^\circ=0.6$ and sin$53.1^\circ=0.8$)

           
1. $69$ cm
2. $72$ cm
3. $79$ cm
4. $65$ cm

To maintain a rotor at a uniform angular speed of $200~\text{rad/s},$ an engine needs to transmit a torque of $180~\text{N-m}.$ What is the power required by the engine?
1.  $33~\text{kW}$
2.  $36~\text{kW}$
3.  $28~\text{kW}$
4.  $76~\text{kW}$