3 The resistive force, $F$, on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, $r$, the velocity of the sphere, $v$, and the viscosity of the fluid, $\eta$. You are given that $\eta$ is measured in $\mathrm { Nm } ^ { - 2 } \mathrm {~s}$.
\begin{enumerate}[label=(\alph*)]
\item By considering its units, find the dimensions of viscosity.

A model of the resistive force suggests the following relationship: $F = 6 \pi \eta ^ { \alpha } r ^ { \beta } v ^ { \gamma }$.
\item Explain whether or not it is possible to use dimensional analysis to verify that the constant $6 \pi$ is correct.
\item Use dimensional analysis to find the values of $\alpha , \beta$ and $\gamma$.

A sphere of radius $r$ and mass $m$ falls vertically from rest through the fluid. After a time $t$ its velocity is $v$.
\item By setting up and solving a differential equation, show that $\mathrm { e } ^ { - k t } = \frac { g - k v } { g }$ where $k = \frac { 6 \pi \eta r } { m }$.

As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.
\item Find, in terms of $g$ and $k$, the terminal velocity of the sphere.

In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.
\item Describe how, according to the model, the terminal velocity of the sphere changes as the viscosity of the fluid through which it falls increases.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q3 [15]}}