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:

1.	\(\vec r\cdot\vec \tau\neq0\text{ and }\vec F\cdot\vec \tau=0\)
2.	\(\vec r\cdot\vec \tau>0\text{ and }\vec F\cdot\vec \tau<0\)
3.	\(\vec r\cdot\vec \tau=0\text{ and }\vec F\cdot\vec \tau=0\)
4.	\(\vec r\cdot\vec \tau=0\text{ and }\vec F\cdot\vec \tau\neq0\)

A wheel has an angular acceleration of \(3.0\) rad/s² and an initial angular speed of \(2.0\) rad/s. In a time of \(2\) s, it has rotated through an angle (in radians) of:

A particle of mass \(m\) moves in the XY plane with a velocity \(v\) along the straight line AB. If the angular momentum of the particle with respect to the origin \(O\) is \(L_A\) when it is at \(A\) and \(L_B\) when it is at \(B,\) then: 
         

A solid cylinder of mass \(50\) kg and radius \(0.5\) m is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely. The tension in the string required to produce an angular acceleration of \(2\) revolutions/s² will be:
1. \(25\) N
2. \(50\) N
3. \(78.5\) N
4. \(157\) N

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:

If \(\theta\) is the angle between two vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$, and $|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}|=\overrightarrow{\mathrm{a}}.\overrightarrow{\mathrm{b}}$, then \(\theta\) is equal to:
1. \(0^\circ\)
2. \(180^\circ\)
3. \(135^\circ\)
4. \(45^\circ\)

A rope is wrapped around a hollow cylinder with a mass of \(3\) kg and a radius of \(40\) cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of \(30\) N?
1. \(0.25 ~\text{rad/s}^2 \)
2. \(25 ~\text{rad/s}^2 \)
3. \(5 ~\text{m/s}^2 \)
4. \(25 ~\text{m/s}^2 \)

The one-quarter sector is cut from a uniform circular disc of radius R. This sector has a mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation will be: 

Five particles of mass \(2\) kg each are attached to the circumference of a circular disc of a radius of \(0.1\) m and negligible mass. The moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane will be:
1. \(1\) kg-m²
2. \(0.1\) kg-m²
3. \(2\) kg-m²
4. \(0.2\) kg-m²