Question

A sphere of mass `M` and radius `R` is falling in a viscous fluid. The terminal velocity attained by the falling object will be alphaortional to :
A. `R^2`
B. `R`
C. `1//R`
D. `1//R^2`

Answer 1

Correct Answer - A
Since sphere is moving with constant velocity, there is no acceleration in it.
When the sphere of radius `R` is falling in a liquid of density `sigma` and coefficient of viscosity `eta` it attains a terminal velocity `v`, under two forces
(i) Effective force acting downward
=`V(rho - sigma) g = (4)/(3) pi R^3 (rho - sigma) g`
where `rho` is density of sphere.
(ii) Viscous forces acting upwards `= 6 pi eta R v`
Since the sphere is moving with a constant velocity `v`, there is no acceleration in it, the net force acting on it must be zero. That is
`6 pi eta Rv = (4)/(3) pi R^3 (rho - sigma)g`
`rArr v = (2)/(9) (R^2 (rho - sigma)g)/(eta)`
=`v prop R^2`
Thus, terminal velocity is alphaortional to the square of its radius.