Vertical circle – surface contact (sphere/track, leaving surface)

6. A smooth sphere, with centre \(O\) and radius \(a\), is fixed with its lowest point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\sqrt { \frac { a g } { 5 } }\) and moves along the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = \frac { a g } { 5 } ( 11 - 10 \cos \theta )\). The particle leaves the surface of the sphere at the point \(C\).
   Find
2. the speed of \(P\) at \(C\) in terms of \(a\) and \(g\),
3. the size of the angle between the floor and the direction of motion of \(P\) at the instant immediately before \(P\) hits the floor.

4 A smooth sphere, with centre \(O\) and radius \(a\), has its lowest point fixed on a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the highest point on the outer surface of the sphere. In the subsequent motion, \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\). Show that, while \(P\) remains in contact with the sphere, the magnitude of the reaction of the sphere on \(P\) is \(m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the surface of the sphere when \(\theta = \alpha\). Given that \(u = \frac { 1 } { 2 } \sqrt { } ( g a )\), find

1. \(\cos \alpha\),
2. the vertical component of the velocity of \(P\) as it strikes the horizontal plane.

3 A particle \(P\), of mass \(m\), is placed at the highest point of a fixed solid smooth sphere with centre \(O\) and radius \(a\). The particle \(P\) is given a horizontal speed \(u\) and it moves in part of a vertical circle, with centre \(O\), on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the surface of the sphere, the speed of \(P\) is \(v\) and the reaction of the sphere on \(P\) has magnitude \(R\). Show that \(R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the sphere at the instant when \(v = 2 u\). Find \(u\) in terms of \(a\) and \(g\).

4 The diagram shows two particles P and Q , of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m . Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to \(\theta ^ { \circ }\). The pulley, P and Q all lie in the same vertical plane. \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-4_499_492_559_251} Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P .

1. Given that P is in equilibrium at the point where \(\theta = \alpha\), determine the value of \(\alpha\). Particle P is now released from rest at the point on the surface where \(\theta = 35\), and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.
2. By considering energy, determine the speed of P at the instant when \(\theta = 45\).
3. State one modelling assumption you have made in determining the answer to part (b).

4. \begin{figure}[h] \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,

• the speed of \(P\) is \(v\)
• the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
  1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
  2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

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

1. 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 \(U = \sqrt { \frac { 2 r g } { 3 } }\)
2. 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.
3. 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.
4. Find the value of \(\alpha\).

7 A smooth hemisphere, of radius \(a\) and centre \(O\), is fixed with its plane face on a horizontal surface. A particle, of mass \(m\), can move freely on the surface of the hemisphere. The particle is placed at the point \(A\), the highest point of the hemisphere, and is set in motion along the surface with speed \(u\).

1. While the particle is in contact with the hemisphere at a point \(P , O P\) makes an angle \(\theta\) with the upward vertical. \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-5_366_1246_715_395} Show that the speed of the particle at \(P\) is $$\left( u ^ { 2 } + 2 g a [ 1 - \cos \theta ] \right) ^ { \frac { 1 } { 2 } }$$
2. The particle leaves the surface of the hemisphere when \(\theta = \alpha\). Find \(\cos \alpha\) in terms of \(a , u\) and \(g\).

A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac{1}{2}u\) the angle \(\theta\) between \(OP\) and the downward vertical satisfies the equation $$8ga(1 - \cos\theta) = 3u^2.$$ [2] Find, in terms of \(m\), \(u\), \(a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position. [4]

\includegraphics{figure_3} A smooth cylinder of radius \(a\) is fixed with its axis horizontal. The point \(O\) is the centre of a circular cross-section of the cylinder. The line \(AOB\) is a diameter of this circular cross-section and the radius \(OA\) makes an angle \(\alpha\) with the upward vertical (see diagram). It is given that \(\cos \alpha = \frac{3}{5}\). A particle \(P\) of mass \(m\) moves on the inner surface of the cylinder in the plane of the cross-section. The particle passes through \(A\) with speed \(u\) along the surface in the downwards direction. The magnitude of the reaction between \(P\) and the inner surface of the sphere is \(R_A\) when \(P\) is at \(A\), and is \(R_B\) when \(P\) is at \(B\). It is given that \(R_B = 10R_A\). Show that \(u^2 = ag\). [6] The particle loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical. Find the value of \(\cos \theta\). [4]

The diagram shows two identical particles, each of mass \(m\) kg, connected by a thin, light inextensible string. \(P\) slides on the surface of a smooth right circular cylinder fixed with its axis, through \(O\), horizontal. \(Q\) moves vertically. \(OP\) makes an angle \(\theta\) radians with the horizontal. \includegraphics{figure_6} The system is released from rest in the position where \(\theta = 0\).

1. Show that the vertical distance moved by \(Q\) is \(\frac{\theta}{\sin \theta}\) times the vertical distance moved by \(P\). [4 marks]
2. In the position where \(\theta = \frac{\pi}{6}\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left(1-\frac{\pi}{6}\right)mg\) N. [9 marks]

A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}

1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]

The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).

1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]