A small sphere of radius \(r\) falls from rest in a viscous liquid. As a result, heat is produced due to the viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to:
1. \(r^3\)
2. \(r^2\)
3. \(r^5\)
4. \(r^4\)

A rectangular film of liquid is extended from \((4~\text{cm} \times 2~\text{cm})\) to \((5~\text{cm} \times 4~\text{cm}).\) If the work done is \(3\times 10^{-4}\) J, the value of the surface tension of the liquid is:
1. \(0.250\) Nm^–1
2. \(0.125\) Nm^–1
3. \(0.2\) Nm^–1
4. \(8.0\) Nm^–1

Three liquids of densities \(\rho_1,\rho_2\) and \(\rho_3\) \((\rho_1>\rho_2>\rho_3)\) having the same value of surface tension \(T\), rise to the same height in three identical capillaries. The angles of contact $\mathrm{}$\(\theta_1\), $\mathrm{}$\(\theta_2\)_, and $\mathrm{}$\(\theta_3\) obey:
1. \( \frac{\pi}{2}>\theta_1>\theta_2>\theta_3 \geq 0 \)
2. \( 0 \leq \theta_1<\theta_2<\theta_3<\frac{\pi}{2} \)
3. \( \frac{\pi}{2}<\theta_1<\theta_2<\theta_3<\pi \)
4. \( \pi>\theta_1>\theta_2>\theta_3>\frac{\pi}{2} \)

A U-tube with both ends open to the atmosphere is partially filled with water. Oil, which is immiscible with water, is poured into one side until it stands at a level of \(10~\text{mm}\) above the water level on the other side. Meanwhile, the water rises by \(65~\text{mm}\) from its original level (see diagram). The density of the oil is: 
              
1. \(425~\text{kg m}^{-3}\)
2. \(800~\text{kg m}^{-3}\)
3. \(928~\text{kg m}^{-3}\)
4. \(650~\text{kg m}^{-3}\)

Two non-mixing liquids of densities \(\rho\) and \(n\rho\) \((n>1)\) are put in a container. The height of each liquid is \(h.\) A solid cylinder of length \(L\) and density \(d\) is put in this container. The cylinder floats with its axis vertical and length \(rL~(r<1))\) in the denser liquid. The density \(d\) is equal to:
1. \([2+(n+1)r ]\rho\)
2. \([2+(n-1)r] \rho\)
3. \([1+(n-1)r] \rho\)
4. \([1+(n+1)r ]\rho\)

The cylindrical tube of a spray pump has a radius \(R,\) one end of which has \(n\) fine holes, each of radius \(r.\) If the speed of the liquid in the tube is \(v,\) the speed of the ejection of the liquid through the holes is:
 

Water rises to height '\(h\)' in a capillary tube. If the length of capillary tube above the surface of the water is made less than \('h'\), then:

The heart of a man pumps \(5~\text{L}\) of blood through the arteries per minute at a pressure of \(150~\text{mm}\) of mercury. If the density of mercury is \(13.6\times10^{3}~\text{kg/m}^{3}\)$$ $\mathrm{and}{\mathrm{}}^{}$ \(g = 10~\text{m/s}^2\), then the power of the heart in watt is:
1. \(1.70\)
2. \(2.35\)
3. \(3.0\)
4. \(1.50\)

The approximate depth of an ocean is \(2700~\text{m}\). The compressibility of water is \(45.4\times10^{-11}~\text{Pa}^{-1}\) and the density of water is \(10^{3}~\text{kg/m}^3\). What fractional compression of water will be obtained at the bottom of the ocean?
1. \(0.8\times 10^{-2}\)
2. \(1.0\times 10^{-2}\)
3. \(1.2\times 10^{-2}\)
4. \(1.4\times 10^{-2}\)

A wind with a speed of \(40\) m/s blows parallel to the roof of a house. The area of the roof is \(250\) m². Assuming that the pressure inside the house is atmospheric pressure, the force exerted by the wind on the roof and the direction of the force will be: (\(\rho_{\text {air }}=1.2\)${\mathrm{ }}_{}\mathrm{kg}/{\mathrm{m}}^3$)
1. \(4 \times 10^5\) N, downwards
2. \(4 \times 10^5\) N, upwards
3. \(2.4 \times 10^5\) N, upwards
4. \(2.4 \times 10^5\) N, downwards