The density of a material in a CGS system of units is \(4~\text{grams/cm}^3\). In a system of units in which the unit of length is \(10~\text{cm}\) and the unit of mass is \(100~\text{grams}\), the value of the density of the material will be:
1. \( 0.04 \)
2. \( 0.4 \)
3. \( 40 \)
4. \(400\)

A particle covers half of its total distance with speed \(v_1\) and the rest half distance with speed \(v_2.\)
Its average speed during the complete journey is:

1. $\frac{v_1+v_2}{2}$

2. $\frac{v_1{v}_2}{v_1+v_2}$

3. $\frac{2v_1{v}_2}{v_1+v_2}$

4. $\frac{v_1^2{v}_2^2}{v_1^2+v_2^2}$

A mass \(m\) moving horizontally (along the x-axis) with velocity \(v\) collides and sticks to a mass of \(3m\) moving vertically upward (along the y-axis) with velocity \(2v.\) The final velocity of the combination is:

A conveyor belt is moving at a constant speed of \(2\) m/s. A box is gently dropped on it. The coefficient of friction between them is \(\mu = 0.5\). The distance that the box will move relative to the belt before coming to rest on it, taking \(g = 10\) ms^–2 is:

A small mass attached to a string rotates on a frictionless table top as shown. If the tension in the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2, the kinetic energy of the mass will:
       
1. decrease by a factor of 2
2. remain constant
3. increase by a factor of 2
4. increase by a factor of 4

A particle of mass \(m\) is thrown upwards from the surface of the earth, with a velocity \(u.\) The mass and the radius of the earth are, respectively, \(M\) and \(R.\) \(G\) is the gravitational constant and \(g\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(u\) so that the particle does not return back to earth is:

1. \(\sqrt{\dfrac{2 {GM}}{{R}^2}} \)

2. \(\sqrt{\dfrac{2 {GM}}{{R}}} \)

3.\(\sqrt{\dfrac{2 {gM}}{{R}^2}} \)

4. \(\sqrt{ {2gR^2}}\)

A particle of mass M is situated at the centre of a spherical shell of the same mass and radius a. The magnitude of the gravitational potential at a point situated at a/2 distance from the centre will be:

1. $-\frac{\mathrm{GM}}{\mathrm{a}}$

2. $-\frac{2\mathrm{GM}}{\mathrm{a}}$

3. $-\frac{3\mathrm{GM}}{\mathrm{a}}$

4. $-\frac{4\mathrm{GM}}{\mathrm{a}}$

A projectile is fired at an angle of \(45^\circ\) with the horizontal. The elevation angle \(\alpha\) of the projectile at its highest point, as seen from the point of projection is:
1. \(60^\circ\)
2. \(tan^{-1}\left ( \frac{1}{2} \right )\)
3. \(tan^{-1}\left ( \frac{\sqrt{3}}{2} \right )\)
4. \(45^\circ\)

At a pressure of \(2\) atmospheres, a mass of diatomic gas \((\gamma = 1.4)\), is compressed adiabatically, causing its temperature to rise from \(27^{\circ}\mathrm{C}\) to \(927^{\circ}\mathrm{C}\). The pressure of the gas in the final state is:
1. 8 atm
2. 28 atm
3. 68.7 atm
4. 256 atm