7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-24_506_640_296_715}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}

A thin smooth hollow spherical shell has centre $O$ and radius $r$. Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point $A$ is the lowest point of the bowl, as shown in Figure 5.

The point $B$ is on the rim of the bowl, with $O B$ at an angle $\theta$ to the upward vertical, where $\tan \theta = \frac { 12 } { 5 }$\\
A small ball is placed in the bowl at $A$. The ball is projected from $A$ with horizontal speed $u$ and moves in the vertical plane $A O B$. The ball stays in contact with the bowl until it reaches $B$.

At the instant when the ball reaches $B$, the speed of the ball is $v$.\\
By modelling the ball as a particle and ignoring air resistance,
\begin{enumerate}[label=(\alph*)]
\item use the principle of conservation of mechanical energy to show that

$$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
\item show that $u ^ { 2 } \geqslant \frac { 41 } { 13 } g r$

The point $C$ is such that $B C$ is a diameter of the rim of the bowl.\\
Given that $u ^ { 2 } = 4 g r$
\item use the model to show that, after leaving the inner surface of the bowl at $B$, the ball falls back into the bowl before reaching $C$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2024 Q7 [13]}}