Particle in hemispherical bowl

4
A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).

1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
2. Find the speed of the particle.
3. Find the time taken for the particle to complete one revolution.

4 \includegraphics[max width=\textwidth, alt={}, center]{727412ec-d783-4392-8b84-e7d5435a3f4e-2_387_613_1073_767} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   1. the angle between \(O P\) and the vertical,
   2. the angular speed of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   1. the angle between \(O P\) and the vertical,
   2. the angular speed of \(P\).

1. \begin{figure}[h] \end{figure} A hemispherical bowl of internal radius \(2 r\) is fixed with its circular rim horizontal. A particle \(P\) is moving in a horizontal circle of radius \(r\) on the smooth inner surface of the bowl, as shown in Figure 1. Particle \(P\) is moving with constant angular speed \(\omega\). Show that \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)

1. \begin{figure}[h] \end{figure} A hemispherical bowl, of internal radius \(r\), is fixed with its circular rim upwards and horizontal. A particle \(P\) of mass \(m\) moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance \(\frac { 1 } { 2 } r\) vertically below the centre of the bowl, as shown in Figure 1.
The time taken by \(P\) to complete one revolution of its circular path is \(T\).
Show that \(T = \pi \sqrt { \frac { 2 r } { g } }\).

2. \begin{figure}[h] \end{figure} A thin hemispherical shell, with centre \(O\) and radius \(a\), is fixed with its open end uppermost and horizontal. A particle \(P\) of mass \(m\) moves in a horizontal circle on the smooth inner surface of the shell. The vertical distance of \(P\) below the level of \(O\) is \(d\), as shown in Figure 2.

1. Find, in terms of \(m , g , d\) and \(a\), the magnitude of the force exerted on \(P\) by the inner surface of the hemisphere. The particle moves with constant speed \(v\).
2. Find \(v\) in terms of \(g , a\) and \(d\).

1. \begin{figure}[h] \end{figure} A hemispherical bowl, of internal radius \(r\), is fixed with its circular rim upwards and horizontal. A particle \(P\) of mass \(m\) moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance \(\frac { 1 } { 2 } r\) vertically below the centre of the bowl, as shown in Figure 1.
The time taken by \(P\) to complete one revolution of its circular path is \(T\).
Show that \(T = \pi \sqrt { \frac { 2 r } { g } }\).

4. A particle \(P\) of mass \(m\) moves on the smooth inner surface of a spherical bowl of internal radius \(r\). The particle moves with constant angular speed in a horizontal circle, which is at a depth \(\frac { 1 } { 2 } r\) below the centre of the bowl.

1. Find the normal reaction of the bowl on \(P\).
2. Find the time for \(P\) to complete one revolution of its circular path.
   (6)
   (Total 10 marks)

1. \begin{figure}[h] \end{figure} A hemispherical bowl of internal radius \(4 r\) is fixed with its circular rim horizontal. The centre of the circular rim is \(O\) and the point \(A\) on the surface of the bowl is vertically below \(O\). A particle \(P\) moves in a horizontal circle, with centre \(C\), on the smooth inner surface of the bowl. The particle moves with constant angular speed \(\sqrt { \frac { 3 g } { 8 r } }\) The point \(C\) lies on \(O A\), as shown in Figure 1.
Find, in terms of \(r\), the distance \(O C\).

7 \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre \(O\) and radius \(5 \mathrm {~m} . P\) moves with constant angular speed \(\omega\) in a horizontal circle at a vertical distance of 3 m below the level of \(O\) (see Fig.1).

1. Calculate the magnitude of the force exerted by the hemisphere on \(P\).
2. Calculate \(\omega\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string is now attached to \(P\). The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. \(P\) moves in the same horizontal circle as before (see Fig. 2).
3. Calculate the new angular speed of \(P\).

10 {} 7

(continued)

\section*{OCR
RECOGNISING ACHIEVEMENT}

2. \begin{figure}[h] \end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\) The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.

1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
2. find \(\omega\) in terms of \(g\) and \(a\).
3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.

1. \begin{figure}[h] \end{figure} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac { 1 } { 4 } a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a hemispherical bowl of internal radius \(10 d\) that is fixed with its circular rim horizontal. The centre of the circular rim is at the point \(O\).
A particle \(P\) moves with constant angular speed on the smooth inner surface of the bowl. The particle \(P\) moves in a horizontal circle with radius \(8 d\) and centre \(C\).

1. Find, in terms of \(g\), the exact magnitude of the acceleration of \(P\). The time for \(P\) to complete one revolution is \(T\).
2. Find \(T\) in terms of \(d\) and \(g\).

A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{8}{3}ga}\). Find \(x\) in terms of \(a\). [6]

\includegraphics{figure_2} A particle \(P\) of mass \(m\) moves on the smooth inner surface of a hemispherical bowl of radius \(r\). The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed \(\sqrt{\left(\frac{3g}{2r}\right)}\) in a horizontal circle at depth \(d\) below the centre of the bowl.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the bowl on \(P\). [4]
2. Find \(d\) in terms of \(r\). [4]

The diagram shows a particle \(P\) of mass \(m\) kg moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r\) m which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h\) m below the top of the bowl. \includegraphics{figure_6}

1. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac{mgr}{h}\) N. [4 marks]
2. Find, in terms of \(g\), \(h\) and \(r\), the constant speed of \(P\). [4 marks]

The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(mg\) applied tangentially as shown. The reaction of the surface of the sphere on \(P\) now has magnitude \(S\) N. \includegraphics{figure_6b}

1. Given that \(r = 2h\), prove that \(S < \frac{1}{6}R\). [5 marks]

\includegraphics{figure_12} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.

1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]