A thin circular ring of mass \(M\) and radius \(r \) is rotating about its axis with constant angular velocity \(ω.\) Two objects each of mass \(m\) are attached gently to the opposite ends of the diameter of the ring. The ring now rotates with angular velocity given by:

1. \(\dfrac{2 M \omega}{M + 2 m}\)

2. \(\dfrac{\left(\right. M + 2 m \left.\right) \omega}{M}\)

3. \(\dfrac{M \omega}{M + 2 m}\)

4. \(\dfrac{\left(\right. M + 2 m \left.\right) \omega}{2 m}\)

From a circular disc of radius \(R\) and mass \(9M,\) a small disc of mass \(M\) and radius \(R/3\) is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is:
1. \(MR^{2}\)
2. \(4MR^{2}\)
3. \(\frac{4}{9}MR^{2}\)
4. \(\frac{40}{9}MR^{2}\)

A particle of mass \(M\) starting from rest undergoes uniform acceleration. If the speed acquired in time \(T\) is \(V\), the power delivered to the particle is:
1. \(\frac{1}{2}\frac{MV^2}{T^2}\)
2. \(\frac{MV^2}{T^2}\)
3. \(\frac{1}{2}\frac{MV^2}{T}\)
4. \(\frac{MV^2}{T}\)

A solid cylinder and a hollow cylinder both of the same mass and same external diameter are released from the same height at the same time on an inclined plane. Both roll down without slipping. Which one will reach the bottom first:

The dependence of acceleration due to gravity \('g'\) on the distance \('r'\) from the centre of the earth, assumed to be a sphere of radius \(R\) of uniform density, is as shown in figure below:

(a)		(b)
(c)		(d)

                    
The correct figure is:
1. \(a\)

2. \(b\)

3. \(c\)

4. \(d\)

The additional kinetic energy to be provided to a satellite of mass \(m\) revolving around a planet of mass \(M,\) to transfer it from a circular orbit of radius \(R_1\) to another of radius \(R_2\) (\(R_2>R_1\)) is:

1. \(GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

2. \(2GmM\) $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

3. $\frac{1}{2}GmM$ $\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

4. \(GmM\) $\left(\frac{1}{R_1^2}-\frac{1}{R_2^2}\right)$

A student measures the distance traversed in free fall of a body, initially at rest in a given time. He uses this data to estimate \(g,\) the acceleration due to gravity. If the maximum percentage errors in the measurement of the distance and the time are \(e_1\) and \(e_2\) respectively, the percentage error in the estimation of \(g\) is:
1. $e_1+2e_2$
2. $e_1+e_2$
3. $e_1-2e_2$
4. $e_2-e_1$

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is:
1. \(15^{\circ}\)
2. \(30^{\circ}\)
3. \(45^{\circ}\)
4. \(60^{\circ}\)

Given the following statements:

Which one of the following pairs of statements is correct?

The electric field of an electromagnetic wave in free space is given by –
$\overrightarrow{E}=10$ $\cos$ $({10}^{7}t+kx)$ $\hat{j}$ $V/m$, where t and x are in seconds and metres respectively. It can be inferred that:

(a) The wavelength λ is 188.4 m
(b) The wave number k is 0.33 rad / m
(c) The wave amplitude is 10 V / m
(d) The wave is propagating along + x direction

Which one of the following pairs of statements is correct?

1. (a) and (b)

2. (b) and (c)

3. (c) and (d)

4. (a) and (c)