What is the increase in pressure required to decrease the volume of water sample by \(0.01\%\)?. (Bulk modulus of water \(=2.1 \times 10^9 \mathrm{~Nm}^{-2}\))
1. \(4.3 \times 10^4 \mathrm{~N} / \mathrm{m}^2 \)
2. \( 1.8 \times 10^7 \mathrm{~N} / \mathrm{m}^2 \)
3. \( 2.1 \times 10^5 \mathrm{~N} / \mathrm{m}^2 \)
4. \(3.7 \times 10^4 \mathrm{~N} / \mathrm{m}^2\)

Water level is maintained in a cylindrical vessel up to a fixed height \(H.\) The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel, so that the water stream coming out of the hole strikes the horizontal plane of the greatest distance from the vessel? 

1. \(h=\frac{H}{2}\)
2. \(h=\frac{3H}{2}\)
3. \(h=\frac{2H}{3}\)
4. \(h=\frac{3}{4}H\)

The figure shows a spring + block + pulley system which is light. The time period of mass would be:

1. \(2\pi\sqrt{\frac{k}{m}}\)
2. \(\frac{1}{2\pi}\sqrt{\frac{k}{m}}\)
3. \(2\pi\sqrt{\frac{m}{k}}\)
4. None of these

A pendant having a bob of mass \(m\) is hanging in a ship sailing along the equator from east to west. When the strip is stationary with respect to water, the tension in the string is \(T_0.\) The difference between \(T_0\) and earth attraction on the bob, is:
1. \(\frac{mg+m\omega^2R}{2}\)
2. \(\frac{m\omega^2R}{3}\)
3. \(\frac{m\omega^2R}{2}\)
4. \(\frac{m\omega^2}R\)

A solid sphere is set into motion on a rough horizontal surface with a linear speed \(v\) in the forward direction and an angular speed \(\frac{v}{R}\) in the anticlockwise direction as shown in the figure. The linear speed of the sphere when it stops rotating is:

1. \(\frac{3v}{5}\)
2. \(\frac{2v}{5}\)
3. \(3v\)
4. \(\frac{6v}{5}\)

Two blocks of masses \(m_1\) and \(m_2\) are connected by a spring of spring constant \(k.\) The block of mass \(m_2\) is given a sharp impulse so that it acquires a velocity \(v_0\) towards the right. What is the maximum elongation that the spring will suffer?

1. \(\left[\frac{\mathrm{m}_1 \mathrm{~m}_2}{\mathrm{~m}_1+\mathrm{m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)
2. \(\left[\frac{\mathrm{m}_1+\mathrm{m}_2}{\mathrm{~m}_1-\mathrm{m}_2}\right] \mathrm{v}_0\)
3. \(\left[\frac{\mathrm{m}_1+\mathrm{m}_2}{\mathrm{~m}_1-\mathrm{m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)
4. \(\left[\frac{2 \mathrm{~m}_1+\mathrm{m}_2}{\mathrm{~m}_1 \mathrm{~m}_2}\right]^{\frac{1}{2}} \mathrm{v}_0\)

A ball of mass \(m\) hits the floor with a speed \(v\) making an angle of incidence \(e\) with the normal. The coefficient of restitution is \(e.\) The speed of the reflected ball and the angle of reflection of the ball will be:

1. \(v'=v,~\theta=\theta'\)
2. \(v'=\frac v2,~\theta=2\theta'\)
3. \(v'=2v,~\theta=2\theta'\)
4. \(v'=\frac{3v}{2},~\theta=\frac{2\theta'}3\)

A particle slides on the surface of a fixed smooth sphere starting from the  topmost point. The angle rotated by the radius through the particle, when it leaves contact with the sphere, is:
1. \(\theta=cos^{-1}\Big(\frac13\Big)\)
2. \(\theta=cos^{-1}\Big(\frac23\Big)\)
3. \(\theta=tan^{-1}\Big(\frac13\Big)\)
4. \(\theta=sin^{-1}\Big(\frac43\Big)\)

What is the radius of curvature of the parabola traced out by the projectile in the previous problem at a point where the particle velocity makes an angle \(\frac{\theta}{2}\) with the horizontal?
1. \(r=\frac{v^2\cos^2\theta}{g\cos^2\theta}\)
2. \(r=\frac{2v\sin\theta}{g\tan\theta}\)
3. \(r=\frac{v\cos\theta}{g\sin^2\frac{\theta}{2}}\)
4. \(r=\frac{3v\cos\theta}{g\cot\theta}\)