Particle on inner surface of sphere/bowl

7. \begin{figure}[h] \end{figure} A hollow sphere has internal radius \(r\) and centre \(O\). A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point \(B\) is the lowest point of the inner surface of the bowl. The point \(A\), where angle \(A O B = 120 ^ { \circ }\), lies on the rim of the bowl, as shown in Figure 4. A particle \(P\) of mass \(m\) is projected from \(A\), with speed \(U\) at \(90 ^ { \circ }\) to \(O A\), and moves on the smooth inner surface of the bowl. The motion of \(P\) takes place in the vertical plane \(O A B\).

1. Find, in terms of \(m , g , U\) and \(r\), the magnitude of the force exerted on \(P\) by the bowl at the instant when \(P\) passes through \(B\).
2. Find, in terms of \(g , U\) and \(r\), the greatest height above the floor reached by \(P\). Given that \(U > \sqrt { 2 g r }\)
3. show that, after leaving the surface of the bowl, \(P\) does not fall back into the bowl.

6. \begin{figure}[h] \end{figure} Figure 5 shows a hollow sphere, with centre \(O\) and internal radius \(a\), which is fixed to a horizontal surface. A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { \frac { 7 a g } { 2 } }\) from the lowest point \(A\) of the inner surface of the sphere. The particle moves in a vertical circle with centre \(O\) on the smooth inner surface of the sphere. The particle passes through the point \(B\), on the inner surface of the sphere, where \(O B\) is horizontal.

1. Find, in terms of \(m\) and \(g\), the normal reaction exerted on \(P\) by the surface of the sphere when \(P\) is at \(B\). The particle leaves the inner surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta , \theta > 0\), with the upward vertical.
2. Show that, after leaving the surface of the sphere at \(C\), the particle is next in contact with the surface at \(A\).

   END

7. \begin{figure}[h] \end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(a\), 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. The point \(B\) is on the rim of the bowl and \(\angle A O B = 120 ^ { \circ }\), as shown in Fig. 4. A smooth small marble of mass \(m\) is placed inside the bowl at \(A\) and given an initial horizontal speed \(u\). The direction of motion of the marble lies in the vertical plane \(A O B\). The marble stays in contact with the bowl until it reaches \(B\). When the marble reaches \(B\), its speed is \(v\).

1. Find an expression for \(v ^ { 2 }\).
2. For the case when \(u ^ { 2 } = 6 g a\), find the normal reaction of the bowl on the marble as the marble reaches \(B\).
3. Find the least possible value of \(u\) for the marble to reach \(B\). The point \(C\) is the other point on the rim of the bowl lying in the vertical plane \(O A B\).
4. Find the value of \(u\) which will enable the marble to leave the bowl at \(B\) and meet it again at the point \(C\).

6. A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(O P\) is inclined at an angle \(\alpha\) to the upward vertical.

1. Show that \(\quad U ^ { 2 } = a g ( 2 + 3 \cos \alpha )\). The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac { 1 } { \sqrt { } 3 }\), (b) show that \(\quad W ^ { 2 } = a g \sqrt { } 3\).

4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\). Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

6 A smooth hemispherical shell of radius \(r \mathrm {~m}\) is held with its circular rim horizontal and uppermost. The centre of the rim is at the point \(O\) and the point on the inner surface directly below \(O\) is \(A\). A small object \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the inner surface of the shell so that \(\angle \mathrm { POA } = \frac { 1 } { 3 } \pi\) radians. At the instant that \(P\) is released, an impulse is applied to \(P\) in the direction of the tangent to the surface at \(P\) in the vertical plane containing \(O , A\) and \(P\). The magnitude of the impulse is denoted by \(I\) Ns.
\(P\) immediately starts to move along the surface towards \(A\) (see diagram).
\(X\) is a point on the circular rim. \(P\) leaves the shell at \(X\).
\includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-5_512_860_829_242} In an initial model of the motion of \(P\) it is assumed that \(P\) experiences no resistance to its motion.

1. Find in terms of \(r , g , m\) and \(I\) an expression for the speed of \(P\) at the instant that it leaves the shell at \(X\).
2. Find in terms of \(r , g , m\) and \(I\) an expression for the maximum height attained by \(P\) above \(X\) after it has left the shell.
3. Find an expression for the maximum mass of \(P\) for which \(P\) still leaves the shell. In a revised model it is assumed that \(P\) experiences a resistive force of constant magnitude \(R\) while it is moving.
4. Show that, in order for \(P\) to still leave the shell at \(X\) under the revised model, $$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
5. Show that the inequality from part (d) is dimensionally consistent.