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 \(C\).

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.

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos \theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos\theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

Answer only one of the following two alternatives. EITHER 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 \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(AOB\) makes an acute angle \(\alpha\) with the vertical, where \(\cos \alpha = \frac{4}{5}\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8 : 9\).

1. Show that \(u^2 = 4ag\). [6]
2. Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere. [6]

OR A machine is used to produce metal rods. When the machine is working efficiently, the lengths, \(x\) cm, of the rods have a normal distribution with mean 150 cm and standard deviation 1.2 cm. The machine is checked regularly by taking random samples of 200 rods. The latest results are shown in the following table.

Interval	\(146 \leqslant x < 147\)	\(147 \leqslant x < 148\)	\(148 \leqslant x < 149\)	\(149 \leqslant x < 150\)
Observed frequency	1	2	23	52

\(150 \leqslant x < 151\)	\(151 \leqslant x < 152\)	\(152 \leqslant x < 153\)	\(153 \leqslant x < 154\)
69	36	15	2

As a first check, the sample is used to calculate an estimate for the mean.

1. Show that an estimate for the mean from this sample is close to 150 cm. [2]

As a second check, the results are tested for goodness of fit of the normal distribution with mean 150 cm and standard deviation 1.2 cm. The relevant expected frequencies, found using the normal distribution function given in the List of Formulae (MF10), are shown in the following table.

Interval	\(x < 147\)	\(147 \leqslant x < 148\)	\(148 \leqslant x < 149\)	\(149 \leqslant x < 150\)
Observed frequency	1	2	23	52
Expected frequency	1.24	8.32	30.94	59.50

\(150 \leqslant x < 151\)	\(151 \leqslant x < 152\)	\(152 \leqslant x < 153\)	\(153 \leqslant x\)
69	36	15	2
59.50	30.94	8.32	1.24

1. Show how the expected frequency for \(151 \leqslant x < 152\) is obtained. [3]
2. Test, at the 5\% significance level, the goodness of fit of the normal distribution to the results. [7]

A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2}ga}\). The particle \(P\) loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\theta = 60°\). [5]
2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\). [5]

\includegraphics{figure_3} Figure 3 shows a fixed hollow sphere of internal radius \(a\) and centre \(O\). A particle \(P\) of mass \(m\) is projected horizontally from the lowest point \(A\) of a sphere with speed \(\sqrt{\left(\frac{5}{4}ag\right)}\). It moves in a vertical circle, centre \(O\), on the smooth inner surface of the sphere. The particle passes through the point \(B\), which is in the same horizontal plane as \(O\). It leaves the surface of the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical.

1. Find, in terms of \(m\) and \(g\), the normal reaction between \(P\) and the surface of the sphere at \(B\). [4]
2. Show that \(\theta = 60°\). [7]

After leaving the surface of the sphere, \(P\) meets it again at the point \(A\).

1. Find, in terms of \(a\) and \(g\), the time \(P\) takes to travel from \(C\) to \(A\). [4]

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 \(OP\) is inclined at an angle \(\alpha\) to the upward vertical.

1. Show that \(U^2 = ag(2 + 3\cos \alpha)\). [7]

The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac{1}{\sqrt{3}}\),

1. show that \(W^2 = ag\sqrt{3}\). [5]

\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).

1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]

\includegraphics{figure_10} A small toy car runs along a track in a vertical plane. The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius \(r\) m. The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram. The car, of mass \(m\) kg, is placed on the track at A, at a height \(h\) m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C. The track is modelled as being smooth, and the car is modelled as a particle P.

1. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is $$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$ where \(\theta\) is the angle between OC and OP. [7]
2. Hence determine, in terms of \(r\), the least possible value of \(h\) so that P can complete a vertical circle. [2]
3. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]

The diagram shows a slide, \(ABC\), at a water park. The shape of the slide may be modelled by two circular arcs, \(AB\) and \(BC\), in the same vertical plane. Arc \(AB\) has radius \(7\) m and subtends an angle \(\alpha\) at its centre \(D\), where \(\cos \alpha = \frac{9}{10}\). Arc \(BC\) has radius \(5\) m and subtends an angle of \(45°\) at its centre, \(O\). The straight line \(DBO\) is vertical. \includegraphics{figure_6} Users of the slide are required to sit in a rubber ring and are released from rest at point \(A\). A girl decides to use the slide. The combined mass of the girl and the rubber ring is \(50\) kg.

1. When the rubber ring is at a point \(P\) on the circular arc \(BC\), its speed is \(v\) ms\(^{-1}\) and \(OP\) makes an angle \(\theta\) with the upward vertical.
   1. Show that \(v^2 = 111.72 - 98\cos\theta\). [4]
   2. Find, in terms of \(\theta\), the reaction between the rubber ring and the slide at \(P\). [4]
   3. Show that, according to this model, the rubber ring loses contact with the slide before reaching \(C\). [3]
   4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii). [1]
2. Show that the rubber ring will remain in contact with the slide along the arc \(AB\). [3]