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 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 distance of 10mm above the water level on the other side. Meanwhile the water rises by 65 mm from its original level (see diagram). The density of the oil is 

(a) $650$ $kg$ $m^{-3}$

(b) $425$ $kg$ $m^{-3}$

(c) $800$ $kg$ $m^{-3}$

(d) $928$ $kg$ $m^{-3}$

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

Three liquids of densities ${\rho }_1,$ ${\rho }_2$ $and$ $$ ${\rho }_3$ (with ${\rho }_1>{\rho }_2>{\rho }_3$), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact ${\theta }_1,{\theta }_2$ $and$ ${\theta }_3$ obey:

 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  floats with its axis vertical and length pL $\left(p<1\right)$ in the denser liquid. The density of the cylinder is d. The density d is equal to:

1. {2+(n+1)p}$\rho$             

2. {2+(n-1)p}$\rho$

3. {1+(n-1)p}$\rho$             

4. {1+(n+1)p}$\rho$

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