7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-24_419_935_251_566}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
The speed of the stone at the instant it reaches \(B\) is \(v\).
The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.
- Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\)
When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\) - show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\)
After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
- Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\).
At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
- Find the value of \(\alpha\).