Question

A solid sphere moves through a liquid. The liquid opposes the motion of the sphere with a force $F$ which depends on the radius $r$, speed $v$ of the correct sphere and coefficient of viscosity $\eta$ of the liquid. The correct relationship between these physical quantities is

• A. $F\propto \eta rv$
• B. $F\propto {\eta }^{-1}rv$
• C. $F\propto \eta r^{-1}v$
• D. $F\propto \eta r^{-1}{v}^2$

Answer 1

The correct option is A $F\propto \eta rv$

$\left[F\right]=k{\left[\eta \right]}^a{\left[r\right]}^b{\left[v\right]}^c$

$ML{T}^{-2}={\left(M{L}^{-1}{T}^{-1}\right)}^a{L}^b{\left(L{T}^{-1}\right)}^c$

$\Rightarrow 1=a;1=-a+b+c,-2=-a-c$

This gives, $a=1,b=1,c=1$

Thus $F=k\eta rv$