Particle on cone surface – no string (normal reaction only)

7 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-4_213_811_260_667} A small ball \(B\) of mass 0.5 kg moves in a horizontal circle with centre \(O\) and radius 0.4 m on the smooth inner surface of a hollow cone fixed with its vertex down. The axis of the cone is vertical and the semi-vertical angle is \(60 ^ { \circ }\) (see diagram).

1. Show that the magnitude of the force exerted by the cone on \(B\) is 5.77 N , correct to 3 significant figures, and calculate the angular speed of \(B\). One end of a light elastic string of natural length 0.45 m and modulus of elasticity 36 N is attached to \(B\). The other end of the string is attached to the point on the axis 0.3 m above \(O\). The ball \(B\) again moves on the surface of the cone in the same horizontal circle as before.
2. Calculate the speed of \(B\).

4. \begin{figure}[h] \end{figure} Figure 3 shows a thin hollow right circular cone fixed with its circular rim horizontal.
The centre of the circular rim is \(O\). The vertex \(V\) of the cone is vertically below \(O\).
The radius of the circular rim is \(4 a\) and \(O V = 3 a\).
A particle \(P\) of mass \(m\) moves in a horizontal circle of radius \(r ( 0 < r < 4 a )\) on the inner surface of the cone. The coefficient of friction between \(P\) and the inner surface of the cone is \(\frac { 1 } { 4 }\) The particle moves with a constant angular speed.
Show that the maximum possible angular speed is \(\sqrt { \frac { 16 g } { 13 r } }\)

1. \begin{figure}[h] \end{figure} A hollow right circular cone, of base radius \(a\) and height \(h\), is fixed with its axis vertical and vertex downwards, as shown in Figure 1. A particle moves with constant speed \(v\) in a horizontal circle of radius \(\frac { 1 } { 3 } a\) on the smooth inner surface of the cone. Show that \(v = \sqrt { } \left( \frac { 1 } { 3 } h g \right)\).

4 A hollow cone is fixed with its axis vertical and its vertex downwards. A small sphere \(P\) of mass \(m \mathrm {~kg}\) is moving in a horizontal circle on the inner surface of the cone. An identical sphere \(Q\) rests in equilibrium inside the cone (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-3_586_611_404_246} The following modelling assumptions are made.

• \(P\) and \(Q\) are modelled as particles.
• The cone is modelled as smooth.
• There is no air resistance.
  1. Assuming that \(P\) moves with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that the total mechanical energy of \(P\) is \(\frac { 3 } { 2 } \mathrm { mv } ^ { 2 } \mathrm {~J}\) more than the total mechanical energy of \(Q\).
  2. Explain how the assumption that \(P\) and \(Q\) are both particles has been used.

In practice, \(P\) will not move indefinitely in a perfectly circular path, but will actually follow an approximately spiral path on the inside surface of the cone until eventually it collides with \(Q\).

Suggest an improvement that could be made to the model.

6 \begin{figure}[h] \end{figure} A container is constructed from a hollow cylindrical shell and a hollow cone which are joined along their circumferences. The cylindrical shell has radius 0.2 m , and the cone has semi-vertical angle \(30 ^ { \circ }\). Two identical small spheres \(P\) and \(Q\) move independently in horizontal circles on the smooth inner surface of the container (see Fig. 1). Each sphere has mass 0.3 kg .

1. \(P\) moves in a circle of radius 0.12 m and is in contact with only the conical part of the container. Calculate the angular speed of \(P\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_278_209_1845_1009} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(Q\) moves with speed \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is in contact with both the cylindrical and conical surfaces of the container (see Fig. 2). Calculate the magnitude of the force which the cylindrical shell exerts on the sphere.
3. Calculate the difference between the mechanical energy of \(P\) and of \(Q\). \section*{[Question 7 is printed overleaf.]}

6. \begin{figure}[h] \end{figure} A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m , is fixed with its axis vertical and its vertex \(V\) pointing downwards, as shown in Figure 4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone. The particle \(P\) moves with constant angular speed \(\omega\) rads \(^ { - 1 }\) The coefficient of friction between the particle \(P\) and the inner surface of the cone is 0.25 Find the greatest possible value of \(\omega\)

2. \begin{figure}[h] \end{figure} A hollow right circular cone, of base diameter \(4 a\) and height \(4 a\) is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle of mass \(m\) moves in a horizontal circle with centre \(C\) on the rough inner surface of the cone with constant angular speed \(\omega\). The height of \(C\) above \(V\) is \(3 a\).
The coefficient of friction between the particle and the inner surface of the cone is \(\frac { 1 } { 4 }\). Find, in terms of \(a\) and \(g\), the greatest possible value of \(\omega\).

\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]

\includegraphics{figure_1} A hollow cone, of base radius \(3a\) and height \(4a\), is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle moves in a horizontal circle with centre \(C\), on the smooth inner surface of the cone with constant angular speed \(\sqrt{\frac{8g}{9a}}\). Find the height of \(C\) above \(V\). [11]