Question

Force of viscosity acting on a spherical body of radius $r$ moving with velocity $v$ through a fluid of viscosity is given by

Answer 1

Step 1: Establish the proportionality relation

Given, viscosity $\eta$ acting on a spherical body of radius $r$ is moving with velocity $v$ through a fluid.

Let $F$ be the force of viscosity. Then,

$F\propto r^a$

$F\propto v^b$

$F\propto {\eta }^c$

$\therefore F\propto r^a{v}^b{\eta }^c$

$\Rightarrow F=k{r}^a{v}^b{\eta }^c$ $\left(1\right)$

Here $k$ is the constant of proportionality.

Step 2: Derive the Formula

Substitute the dimensional formula of each quantity in equation $\left(1\right)$

$\left[M^1{L}^1{T}^{-2}\right]={\left[M^0{L}^1{T}^0\right]}^a{\left[M^0{L}^1{T}^{-1}\right]}^b{\left[M^1{L}^{-1}{T}^{-1}\right]}^c$

$\Rightarrow \left[M^1{L}^1{T}^{-2}\right]=\left[M^c{L}^{a+b-c}{T}^{-b-c}\right]$

Equating the power of $M,L,T$ on both sides,

$c=1$

$a+b-c=1$

$-b-c=-2$

Upon solving the above equations we get,

$a=1,b=1,c=1$

Substituting this equation $\left(1\right)$ we get,

$F=krv\eta$

Therefore, the force of viscosity is $F=krv\eta$.