Question

Obtain an expression for the terminal velocity of a small sphere of radius 'a' (made from a material of density P), falling vertically downwards through a viscous medium of density and coefficient of viscosity n.​

Answer 1

Heya,

This question can be answered using a simple technique.

For this, Let us assume ñ denotes coefficient of viscosity.

Now, What is terminal velocity ?

The constant velocity with which a liquid moves through a fluid under the application of gravitational, viscous and buoyant force is called terminal velocity.

Let the density of the body be P and that of the fluid be d.

The free body diagram of the body is attached hereby.

Now, According to Stoke's law

The viscous force that acts on a spherical body of radius r is given by 6πñrv. (v = velocity )

velocity will be constant when net force will be zero.

I.e. Buoyant force + viscous force = gravitational force.

$d \times \frac{4}{3} \pi {r}^{3} g + 6\pi \: nrv = \frac{4}{3} \pi \: {r}^{3 \: } pg$

Now, solving the above equation, we will get :-

$velocity \: = \frac{2 {r}^{2}(p - d)g }{9n}$

Regards

Kshitij