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 of force is proportional to
Hint: $\left[\eta \right]=\left[M{L}^{-1}{T}^{-1}\right]$

• A. $\frac{\eta r}{v}$
• B. $\eta rv$
• C. ${\eta }^{2}rv$
• D. ${\eta }^{2}r{v}^2$

Answer 1

The correct option is B $\eta rv$

As per the question the viscous force is function of radius, velocity and viscosity.
So, we can write
$F\propto {\eta }^a{r}^b{v}^c$
$\Rightarrow \left[F\right]=\left[\eta {]}^a\right[r{]}^b[v{]}^c$
$\Rightarrow \left[ML{T}^{-2}\right]=\left[M{L}^{-1}{T}^{-1}{]}^a\right[L{]}^b[L{T}^{-1}{]}^c$
$\Rightarrow \left[ML{T}^{-2}\right]=\left[M^a{L}^{(-a+b+c)}{T}^{(-a-c)}\right]$
Equating the exponents of similar quantities on both sides we get $a=1,b=1,c=1$
Thus, the force will be $F\propto \eta rv$