Physics And Measurement Question 259
Question: When a small sphere moves at low speed through a fluid, the viscous force F, opposing the motion is experimentally found to depend upon the radius r, the velocity v of the sphere and the viscosity $ \eta $ of the fluid. Expression for force is
Options:
A) $ 4\pi \eta rv^{2} $
B) $ 4\pi \eta r^{2}v $
C) $ 2\pi \eta r^{2}v $
D) $ 6\pi \eta rv $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] We can thus say that the viscous force (F) is the function of radius (r), velocity (v) and viscosity ( $ \eta $ ). $ orF=f(\eta ,r,v)orF=k{{\eta }^{x}}r^{y}v^{z} $ …… (1)
Where k is a constant. Now, dimensions of the constituents are
$ \therefore [ML{{T}^{-2}}]={{[M{{L}^{-1}}{{T}^{-1}}]}^{x}}{{[L]}^{y}}{{[L{{T}^{-1}}]}^{z}} $
$ =[M^{x}{{L}^{-x+y+z}}{{T}^{-x-z}}] $
Equating the exponents of similar quantities of both sides we get, $ x=1;-x+y+z=1 $ and $ -x-z=-2 $
Solving for $ x,y\And z, $ , we get $ x=y=z=1 $ Equation (1) becomes $ F=k\eta rv $
Experimentally, it was found that $ k=6\pi orF=6\pi \eta rv $ , which is the famous Stokes’ law.
