Question

State and explain Stoke’s law. Explain its applications.

Answer 1

When a body falls in a viscous medium, it carries layers of fluid which are in body’s contact whereas the layers of fluid in contact with the stationary surface remain almost at rest. The layers of fluid destroys the relative motion and motion of the body is thus opposed. The viscous drag increases with velocity of the body till viscous drag and upthrust of the body are together equal to the weight of the body which acts downwards. When there is no net force, the body moves with the uniform velocity. This velocity is called terminal velocity. Stoke showed that the retarding force F due to viscous drag for a spherical body of radius r that moves with a velocity u in a fluid, with coefficient of viscosity η, is given as:

F = 6πηrv

This expression is known as Stoke's Law.

Derivation with help of dimensions

F ∝ v, velocity;

F ∝ r, radius of the body;

F ∝ η, coefficient of viscosity of the fluid.

⇒ F = A η^a r^b v^c

where A is a constant with no dimensions. Putting the dimensions,

[MLT^-2] = [ML^-1T^-1]^a [L]^b [LT^-1]^c

OR [MLT^-2] = [M^aL^-a+b+c T^-a-c]

Comparing the two sides, we get

a = 1, b = 1 and c = 1

∴ F = k ηr v

As per experiemented results, k = 6π

∴ F = 6π ηr v

Terminal Velocity

The constant speed that afrelly object eventually reaches when the resistance of medium thorugh which it is falling prevent further acceleration expression for teminal velocity.

Let ρ be the density of the body and σ be the density of the medium, then,

Weight of body = \(\frac{4}{3}\) πr³ ρg .........(1)

This is the formula for terminal velocity.

It is apparent from the above expression that the terminal velocity is:

(a) Directly proportional to the square of the radius of body.

(b) Directly proportinal t the densities of the body and the medium.

(c) Inversely proportional to the coefficient of viscosity of the medium.

Graph between velocity vs time:

Variation of viscosity with temperature and pressure: (i) Effect of temperature on viscosity:

(i) When a liquid is heated then the kinetic energy of it’s molecules increases and the intermolecular force between them decreases. Hence the viscosity of a liquid decreases with the increase in it’s temperature.

(ii) Viscosity of gases is due to the diffusion of molecules from one moving layer to another. But the rate of diffusion of a gas is directly proportional to the square root of the temperature. So the viscosity of a gas increases with its temperature.

(ii) Effect of pressure:

(i) Except the water, the viscosity of liquid increases with the increase in pressure. In case of water, the viscosity decreases with increase in pressure.

(ii) The viscosity of gas is independent of pressure.