Particle on sphere or circular surface

7. \begin{figure}[h] \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,

1. use the principle of conservation of mechanical energy to show that $$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
2. 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 \(u ^ { 2 } = 4 g r\)
3. 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 \(C\).

5. \begin{figure}[h] \end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(r\), forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at \(A\) with the rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\), where \(\angle A O B = \alpha\) and \(\tan \alpha = \frac { 3 } { 4 }\), is on the rim of the bowl, as shown in Figure 2. A small smooth marble \(M\) is placed inside the bowl at \(A\), and given an initial horizontal speed \(\sqrt { } ( g r )\). The motion of \(M\) takes place in the vertical plane \(O A B\).

1. Show that the speed of \(M\) as it reaches \(B\) is \(\sqrt { } \left( \frac { 3 } { 5 } g r \right)\). After leaving the surface of the bowl at \(B , M\) moves freely under gravity and first strikes the horizontal surface at the point \(C\). Given that \(r = 0.4 \mathrm {~m}\),
2. find the distance \(A C\).

7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre \(O\) and radius \(a\). The point \(A\) is the highest point on the hemisphere. A particle \(P\) is placed on the hemisphere at \(A\). It is then given an initial horizontal speed \(u\), where \(u ^ { 2 } = \frac { 1 } { 2 } ( a g )\). When \(O P\) makes an angle \(\theta\) with \(O A\), and while \(P\) remains on the hemisphere, the speed of \(P\) is \(v\).

1. Find an expression for \(v ^ { 2 }\).
2. Show that, when \(\theta = \arccos 0.9 , P\) is still on the hemisphere.
3. Find the value of \(\cos \theta\) when \(P\) leaves the hemisphere.
4. Find the value of \(v\) when \(P\) leaves the hemisphere. After leaving the hemisphere \(P\) strikes the table at \(B\).
5. Find the speed of \(P\) at \(B\).
6. Find the angle at which \(P\) strikes the table. \section*{Alternative Question 2:}
   1. Two light elastic strings \(A B\) and \(B C\) are joined at \(B\). The string \(A B\) has natural length 1 m and modulus of elasticity 15 N . The string \(B C\) has natural length 1.2 m and modulus of elasticity 30 N . The ends \(A\) and \(C\) are attached to fixed points 3 m apart and the strings rest in equilibrium with \(A B C\) in a straight line.
   Find the tension in the combined string \(A C\).

3 A fixed hollow sphere with centre \(O\) has a smooth inner surface of radius \(a\). A particle \(P\) of mass \(m\) is projected horizontally with speed \(2 \sqrt { } ( a g )\) from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical.

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height that \(P\) reaches above the level of \(O\).

7 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-4_415_1007_1573_513}

1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
2. Find the value of \(\theta\) when the particle leaves the hemisphere.