An engine pumps water continuously through a hose. Water leaves the hose with a velocity \(v\) and \(m\) is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?

1. \(\dfrac{1}{2} m v^{3}\)                                   

2. \(m v^{3}\)

3. \(\dfrac{1}{2} m v^{2}\)                                   

4. \(\dfrac{1}{2} m^{2} v^{2}\)

Two bodies are in equilibrium when suspended in water from the arms of a balance. The mass of one body is \(36~\text g\) and its density is \(9~\text{g/cm}^3.\) If the mass of the other is \(48~\text g,\) its density in \((\text{g/cm}^3)\) will be:
1. \(\frac{4}{3}\)
2. \(\frac{3}{2}\)
3. \(3\)
4. \(5\)

A vertical \(\mathrm{U}\)-tube of uniform inner cross-section contains mercury in both its arms. A glycerin (density\(=1.3\) g/cm³) column of length \(10\) cm is introduced into one of its arms. Oil of density \(0.8\) g/cm³  is poured into the other arm until the upper surfaces of the oil and glycerin are at the same horizontal level. The length of the oil column is:
(density of mercury \(=13.6\) g/cm³)
                       
1. \(10.4\) cm
2. \(8.2\) cm
3. \(7.2\) cm
4. \(9.6\) cm

A triangular lamina of area A and height h is immersed in a liquid of density $\mathrm{\rho }$ in a vertical plane with its base on the surface of the liquid. The thrust on the lamina is:
1. $\frac{1}{2}\mathrm{A\rho gh}$                 

2. $\frac{1}{3}\mathrm{A\rho gh}$

3. $\frac{1}{6}\mathrm{A\rho gh}$                 

4. $\frac{2}{3}\mathrm{A\rho gh}$

For the figures given below, the correct observation is:
              

An iceberg of density \(900 ~\text{kg/m}^ 3\)${\mathrm{}}^{}$ is floating in the water of density \(1000 ~\text{kg/m}^ 3.\) The percentage of the volume of ice cube outside the water is: 
1. \(20\%      \)                                 
2. \(35\%      \)
3. \(10\%      \)                                    
4. \(25\%      \)

A hemispherical bowl just floats without sinking in a liquid of density $1.2\times {10}^3$ $\mathrm{kg}/{\mathrm{m}}^3$. If the outer diameter and the density of the material of the bowl are 1 m and $2\times {10}^4$ $\mathrm{kg}/{\mathrm{m}}^3$ respectively, then the inner diameter of the bowl will be:
1. 0.94 m
2. 0.97 m
3. 0.98 m
4. 0.99 m

A concrete sphere of radius R has a cavity of radius r which is packed with sawdust. The specific gravities of concrete and sawdust are respectively 2.4 and 0.3. For this sphere to float with its entire volume submerged underwater, the ratio of mass of concrete to mass of sawdust will be:

A solid sphere of density $\mathrm{\eta }$ ( > 1) times lighter than water is suspended in a water tank by a string tied to its base as shown in fig. If the mass of the sphere is m, then the tension in the string is given by:

1. $\left(\frac{\mathrm{\eta }-1}{\mathrm{\eta }}\right)\mathrm{mg}$                              

2. $\mathrm{\eta mg}$

3. $\frac{mg}{\mathrm{\eta }}$                                       

4. $(\mathrm{\eta }-1)\mathrm{mg}$

A boat carrying steel balls is floating on the surface of water in a tank. If the balls are thrown into the tank one by one, how will it affect the level of water?