A man (mass = \(50\) kg) and his son (mass = \(20\) kg) are standing on a frictionless surface facing each other. The man pushes his son so that he starts moving at a speed of \(0.70~\text{ms}^{-1}\) with respect to the man. The speed of the man with respect to the surface is:
1. \(0.28~\text{ms}^{-1}\)
2. \(0.47~\text{ms}^{-1}\)
3. \(0.20~\text{ms}^{-1}\)
4. \(0.14~\text{ms}^{-1}\)

A thin smooth rod of length \(L\) and mass \(M\) is rotating freely with angular speed ${}_{}$\(\omega_0\) about an axis perpendicular to the rod and passing through its center. Two beads of mass \(m\) and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be:

The moment of inertia of a body about a given axis is \(1.5\text{ kg.m}^2.\) Initially, the body is at rest. In order to produce rotational kinetic energy of \(1200\text{ J},\) the angular acceleration of \(20\text{ rad/s}^2\) must be applied about the axis for a duration of:

A uniform cylinder of mass \(M\) and radius \(R\) is to be pulled over a step of height \(a\) (\(a<R\)) by applying a force \(F\) at its centre '\(O\)' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of \(F\) required is :

        
1. \( M g \sqrt{1-\frac{a^2}{R^2}} \)
2. \( M g \sqrt{\left(\frac{R}{R-a}\right)^2-1} \)
3. \(M g \frac{a}{R} \)
4. \( M g \sqrt{1-\left(\frac{R-a}{R}\right)^2} \)

Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are \(0.1\) kg-m² and \(10\) rad s^–1 respectively while those for the second one are \(0.2\) kg-m² and \(5\) rad s^–1 respectively. At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is:
1. \(\frac{10}{3}~\text{J}\)
2. \(\frac{2}{3}~\text{J}\)
3. \(\frac{5}{3}~\text{J}\)
4. \(\frac{20}{3}~\text{J}\)

A square shaped hole of side \(l=\dfrac{a}{2}\) is carved out at a distance \(d=\dfrac{a}{2}\) from the centre '\(O\)' of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\dfrac{a}{x}\) value of \(x\) (to the nearest integer) is:

       
1. \(12\)
2. \(23\)
3. \(45\)
4. \(76\)

Moment of inertia of a cylinder of mass \(M\), length \(L\) and radius \(R\) about an axis passing through its centre and perpendicularto the axis of the cylinder is \(I=M\left(\frac{R^2}{4}+\frac{L^2}{12}\right) \) .If such a cylinder is to be made for a given mass of a material, the ratio \(L/R\) for it to have minimum possible \(I\) is:
1. \( \frac{2}{3} \)
2. \( \frac{3}{2} \)
3. \(\sqrt{\frac{2}{3}} \)
4. \( \sqrt{\frac{3}{2}}\)

A person with a mass of \(80~\text{kg}\) is standing on the rim of a circular platform with a mass of \(200~\text{kg}\) and rotating about its axis at a speed of \(5\) revolutions per minute (rpm). As the person moves toward the centre of the platform, what will be the platform's new rotational speed (in rpm) once the person reaches its centre?
1. \(3\)
2. \(6\)
3. \(9\)
4. \(12\)

An massless equilateral triangle \(EFG\) of side \(a\) (As shown in figure) has three particles of mass \(m\) situated at its vertices. The moment of inertia of the system about the line \(EX\) perpendicular to \(EG\) in the plane of \(EFG\) is \(\frac{N}{20} m a^2\) where \(N\) is an integer. The value of \(N\) is:

 
1. \(10\)
2. \(15\)
3. \(20\)
4. \(25\)