3 Solid toy aeroplane nose cones of various sizes are made in the shape shown in Fig. 3.1, where OA is its line of symmetry.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-3_364_432_395_845}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{figure}

The air resistance against the nose cone as the aeroplane flies through the air is initially modelled by $R = k r v \eta$, where $R$ is the air resistance, $r$ is the radius of the circular flat end of the nose cone, $v$ is the velocity of the nose cone, $\eta$ is the viscosity of the air and $k$ is a dimensionless constant.
\begin{enumerate}[label=(\roman*)]
\item Use dimensional analysis to show that the dimensions of $\eta$ are $\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }$.

In an experiment conducted on a particular nose cone, measurements of air resistance are taken for different velocities. The viscosity of the air does not vary during the experiment. The graph in Fig. 3.2 shows the results. Measurements are given using the appropriate S.I. units.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{be1851d6-af11-40e1-8a36-5938ee7864d4-3_794_1166_1411_427}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{figure}
\item Comment on whether the results of this experiment are consistent with the initial model.

It is now suggested that a better model for the air resistance is $R = K r v \left( \frac { \rho r v } { \eta } \right) ^ { \alpha }$, where $\rho$ is the density of the air, $K$ is a dimensionless constant and $R , r , v$ and $\eta$ are as before.
\item (A) Find the dimensions of $\frac { \rho r v } { \eta }$.\\
(B) Explain why you cannot use dimensional analysis to find the value of $\alpha$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS  Q3 [9]}}