Question

A small sphere falls from rest under gravity in a viscous medium, producing heat due to friction. Find how rate of production of heat does depend upon the radius of the sphere at terminal velocity.

Solution

Correct option is

 

The terminal velocity attained by a small sphere of radius r and density  falling in a medium of density  and viscosity  is given by

         

By Stokes’ law, the viscous (frictional) force acting on the sphere at terminal velocity v is given by    

         

The rate of production of heat due to force F is  

         

Substituting the value of v2 from eq. (i), we get   

          

                  

.

Thus, the rate of production of heat is directly proportional to the fifth power of the radius.

SIMILAR QUESTIONS

Q1

The relative velocity between two layers of water is 8.0 cm/s. If the perpendicular distance between the layers is 0.1 cm, find the velocity gradient.

Q2

There is 1-mm thick layer of glycerine between a flat plate of the area 100 cm2 and a big plate. If the coefficient of viscosity of glycerine is 1.0 kg/(m-s), then how much force is required to move the plate with a velocity of 7.0 cm/s?   

Q3

Calculate the terminal velocity of an oil drop falling freely in air. The radius of the drop is 0.01 mm and the density of oil is . The coefficient of viscosity of air is . The density of air is negligible. 

Q4

The critical velocity of an oil drop in air is . What is the radius of the drop? If two such drops coalesce than what will be the terminal velocity of the resultant drop. Coefficient of viscosity of oil is  and density is . The density of air in comparison to oil is negligible and .

Q5

A metal sphere of diameter  enters water after falling a distance h freely in the gravitational field of the earth. After entrance in water, its velocity remains unchanged. Calculate the value of h. The coefficient of viscosity of water , density of water  and acceleration due to gravity = 10 N/kg.     

Q6

An air bubble of radius 1.0 mm rises with uniform velocity through a viscous liquid of density 1625 kg/m3. Calculate the velocity of the bubble if the coefficient of viscosity of the liquid is 10 poise and the density of air is negligible. (g = 10 m/s2). 

Q7

An air bubble of radius 0.1 mm is moving upwards in water with a velocity of 0.35 cm/s. If the density of water is  and gravitational acceleration is 9.8 m/s2 and the density of the air is negligible, then find out the coefficient of viscosity of water.     

Q8

Water at a pressure of  flows at 2.0 m/s through a pipe of 0.02 m2 cross-sectional area which reduces to 0.01 m2. What is the pressure in the smaller in the cross-section of the pipe?

Q9

Water is flowing through two horizontal pipes of different diameters which are connected together. In the first pipe the speed of water is 4.0 m/s and the pressure is . Calculate the speed and pressure of water in the second pipe. The diameters of the pipes are 3.0 cm and 6.0 cm respectively.

Q10

A liquid is kept in a cylindrical vessel which is being rotated about its axis. The liquid rises at the sides. If the radius of the vessel is 0.05 m and the speed of rotation is 2 rev/s, find the difference in the heights of liquid at the centre of the vessel and at its sides.