7. One end of a light rod of length $\frac { 5 } { 7 } \mathrm {~m}$ is attached to a fixed point $O$ and the other end is attached to a particle $P$, of mass $m \mathrm {~kg}$. The particle $P$ is projected from the point $A$, which is vertically below $O$, with a horizontal speed of $u \mathrm {~ms} ^ { - 1 }$ so that it moves in a vertical circle with centre $O$. When the rod $O P$ is inclined at an angle $\theta$ to the downward vertical, the speed of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and the tension in the rod is $T \mathrm {~N}$.\\
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}
\begin{enumerate}[label=(\alph*)]
\item Show that

$$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
\item Hence determine the least possible value of $u ^ { 2 }$ for the particle to reach the highest point of the circle.
\item Given that $u ^ { 2 } = 32 \cdot 2$,
\begin{enumerate}[label=(\roman*)]
\item find, in terms of $m$ and $\theta$, an expression for $T$,
\item calculate the range of values of $\theta$ such that the rod is exerting a thrust.\\

State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer.

Additional page, if required.

Write the question number(s) in the left-hand margin. only
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 3 2024 Q7 [15]}}