Question

A small metal sphere of radius a is falling with a velocity v through a vertical column of a viscous liquid. If the coefficient of viscosity of the liquid is $\eta $, then the sphere encounters an opposing force
A. $6\pi \eta {{a}^{2}}v$
B. $\dfrac{6\eta v}{\pi {{a}^{2}}}$
C. $6\pi \eta av$
D. $\dfrac{\pi \eta v}{6{{a}^{3}}}$

Answer 1

Hint:Liquids exert a force on the bodies moving in them such that the force opposes the motion of the body. This opposing force is called the viscous force. According, to the Stokes law, when a sphere of radius a is moving a liquid with a velocity of v, the viscous force on the sphere is directly proportional to the velocity of the body and the radius of the body.

Complete step by step answer:
We may have observed that when we move our hand in water we feel some restriction in moving our hand. We feel as if some force opposes the motion of our hand. Liquids exert a force on the bodies moving in them such that the force opposes the motion of the body. This opposing force is called the viscous force.

We can see that the tendency of a liquid to keep the body at rest, relative to it. That is why it applies a force that opposes the motion of the body. Now, if we throw a small metallic sphere in a liquid, then the liquid will apply a viscous force on it. The viscous force on the sphere is given by the stroke’s law.

According, to the Stokes law, when a sphere of radius a is moving a liquid with a velocity of v, the viscous force on the sphere is directly proportional to the velocity of the body and the radius of the body. The viscous force also depends on something known as coefficient of viscosity of the liquid ($\eta $). The viscous force is directly proportional to $\eta $.Therefore, the viscous force on the sphere is given as $6\pi \eta av$.

Hence, the correct option is C.

Note:The viscous exerted by a liquid only depends on the radius and the speed of the sphere and the liquid in which it is moving.This means that the viscous force on sphere does not dependent on the gravitational force, which acts in the downward direction.If you do not know about stroke’ law but do know the dimension of $\eta $, the you can simply check which of the options have the same dimension as that of force because we can see that the dimensions of all the options are different.