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 \( \text m\) is suspended from the rod at \(160~\text{cm}\) mark as shown in the figure. What would be the value of \( \mathrm{m}\) such that the rod is in equilibrium? (Take \(g=10~( \text {m/s}^2)\)

A rod of weight \(w\) is supported by two parallel knife edges, A and B, and is in equilibrium in a horizontal position. The knives are at a distance \(d\) from each other. The centre of mass of the rod is at a distance \(x \) from A. The normal reaction on A is:

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 \(200\) rad s^-1, an engine needs to transmit a torque of \(180\) N-m. What is the power required by the engine?
1. \(33\) kW
2. \(36\) kW
3. \(28\) kW
4. \(76\) kW