Three particles of masses \(100~\text{g}\), \(150~\text{g}\), and \(200~\text{g}\) respectively are placed at the vertices of an equilateral triangle of a side \(0.5~\text{m}\) long. What is the position of the centre of mass of three particles?

  

1.	\(\left(\dfrac{5}{18} ,   \dfrac{1}{3 \sqrt{3}}\right) \)	2.	\(\left(\dfrac{1}{4} ,   0\right) \)
3.	\(\left(0 ,   \dfrac{1}{4}\right) \)	4.	\(\left(\dfrac{1}{3 \sqrt{3}} ,   \dfrac{5}{18}\right) \)

A circular disc of radius \(b\) has a hole of radius \(a\) at its centre (see figure). If the mass per unit area of the disc varies as \(\left(\frac{\sigma_0}{r}\right)\), then the radius of gyration of the disc about its axis passing through the centre is:

                             
1. \(\frac{a+b}{3}\)
2. \(\sqrt{\frac{a^2+b^2+ab}{3}}\)
3. \(\sqrt{\frac{a^2+b^2+ab}{2}}\)
4. \(\frac{a+b}{2}\)

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. \( \sqrt{\frac{3}{2}}\)