6.05c Horizontal circles: conical pendulum, banked tracks

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 \(T = \pi \sqrt { \frac { 2 r } { g } }\).

6. The path followed by a motorcycle round a circular race track is modelled as a horizontal circle of radius 50 m . The track is banked at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The motorcycle travels round the track at constant speed. The motorcycle is modelled as a particle and air resistance can be ignored. In an initial model it is assumed that there is no sideways friction between the motorcycle tyres and the track.

1. Find the speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), of the motorcycle. In a refined model it is assumed that there is sideways friction. The coefficient of friction between the motorcycle tyres and the track is \(\frac { 1 } { 4 }\). It is still assumed that air resistance can be ignored and that the motorcycle is modelled as a particle. The motorcycle's path is unchanged. Using this model,
2. find the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at which the motorcycle can travel without slipping sideways.

2. \begin{figure}[h] \end{figure} Figure 2 shows a fairground ride that consists of a chair of mass \(m\) attached to one end of a rigid arm of length \(\frac { 5 a } { 4 }\). The other end of the arm is freely hinged to the rim of a thin horizontal circular disc of radius \(a\). The disc rotates with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. As the ride rotates the arm remains in a vertical plane through the centre of the disc. The arm makes a constant angle \(\theta\) with the vertical, where \(\tan \theta = \frac { 3 } { 4 }\) The chair is modelled as a particle and the arm is modelled as a light rod.

1. Find the tension in the arm in terms of \(m\) and \(g\)
2. Find \(\omega\) in terms of \(a\) and \(g\)

2. \begin{figure}[h] \end{figure} A hemispherical bowl of internal radius \(6 r\) 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 \(\sqrt { \frac { g } { 4 r } }\). The point \(C\) lies on \(O A\), as shown in Figure 1. Find, in terms of \(r\), the distance \(O C\)

4. \begin{figure}[h] \end{figure} A car is travelling round a circular track. The track is banked at an angle \(\alpha\) to the horizontal, as shown in Figure 4. The car and driver are modelled as a particle.
The car moves round the track with constant speed in a horizontal circle of radius \(r\).
When the car is moving with speed \(\frac { 1 } { 2 } \sqrt { g r }\) round the circle, there is no sideways friction between the tyres of the car and the track.

1. Show that \(\tan \alpha = \frac { 1 } { 4 }\) The sideways friction between the tyres of the car and the track has coefficient of friction \(\mu\), where \(\mu < 4\) The maximum speed at which the car can move round the circle without slipping sideways is \(V\).
2. Find \(V\) in terms of \(\mu , r\) and \(g\).

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 \(d\).

5. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a thin smooth fixed vertical pole. One end of a light inextensible string of length \(2 l\) is attached to a point \(A\) on the pole. The other end of the string is attached to \(R\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle \(P\) moves with constant angular speed in a horizontal circle, with both halves of the string taut, and \(A R = \frac { 6 l } { 5 }\), as shown in Figure 2. It may be assumed that in this motion the string does not wrap itself around the pole and that at any instant, the triangle \(A P R\) lies in a vertical plane.

1. Show that the tension in the lower half of the string is \(\frac { 5 m g } { 3 }\)
2. Find, in terms of \(l\) and \(g\), the time for \(P\) to complete one revolution.

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 \(T = \pi \sqrt { \frac { 2 r } { g } }\).

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

2. \begin{figure}[h] \end{figure} A light inextensible string of length \(8 l\) has its ends fixed to two points \(A\) and \(B\), where \(A\) is vertically above \(B\). A small smooth ring of mass \(m\) is threaded on the string. The ring is moving with constant speed in a horizontal circle with centre \(B\) and radius 3l, as shown in Fig. 2. Find

1. the tension in the string,
2. the speed of the ring.
3. State briefly in what way your solution might no longer be valid if the ring were firmly attached to the string.
   (1) \section*{3.} \section*{Figure 3}
   \includegraphics[max width=\textwidth, alt={}]{044c5866-0a12-4309-8ced-b463e1615fb0-3_564_1051_438_541}
   A child's toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig. 3. The cylinder and the hemisphere each have radius \(r\), and the height of the cylinder is \(h\). The material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the cylinder vertical.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is fixed to a point \(A\) which is vertically above the point \(O\) on a smooth horizontal table. The particle \(P\) remains in contact with the surface of the table and moves in a circle with centre \(O\) and with angular speed \(\sqrt { \left( \frac { k g } { 3 a } \right) }\), where \(k\) is a constant. Throughout the motion the string remains taut and \(\angle A P O = 30 ^ { \circ }\), as shown in Figure 3.

1. Show that the tension in the string is \(\frac { 2 k m g } { 3 }\).
2. Find, in terms of \(m , g\) and \(k\), the normal reaction between \(P\) and the table.
3. Deduce the range of possible values of \(k\). The angular speed of \(P\) is changed to \(\sqrt { \left( \frac { 2 g } { a } \right) }\). The particle \(P\) now moves in a horizontal circle above the table. The centre of this circle is \(X\).
4. Show that \(X\) is the mid-point of \(O A\).

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

5. A car of mass \(m\) moves in a circular path of radius 75 m round a bend in a road. The maximum speed at which it can move without slipping sideways on the road is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that this section of the road is horizontal,

1. show that the coefficient of friction between the car and the road is 0.6 . The car comes to another bend in the road. The car's path now forms an arc of a horizontal circle of radius 44 m . The road is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between the car and the road is again 0.6. The car moves at its maximum speed without slipping sideways.
2. Find, as a multiple of \(m g\), the normal reaction between the car and road as the car moves round this bend.
3. Find the speed of the car as it goes round this bend.

3. A rough disc rotates about its centre in a horizontal plane with constant angular speed 80 revolutions per minute. A particle \(P\) lies on the disc at a distance 8 cm from the centre of the disc. The coefficient of friction between \(P\) and the disc is \(\mu\). Given that \(P\) remains at rest relative to the disc, find the least possible value of \(\mu\).

6. A bend of a race track is modelled as an arc of a horizontal circle of radius 120 m . The track is not banked at the bend. The maximum speed at which a motorcycle can be ridden round the bend without slipping sideways is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorcycle and its rider are modelled as a particle and air resistance is assumed to be negligible.

1. Show that the coefficient of friction between the motorcycle and the track is \(\frac { 2 } { 3 }\). The bend is now reconstructed so that the track is banked at an angle \(\alpha\) to the horizontal. The maximum speed at which the motorcycle can now be ridden round the bend without slipping sideways is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the bend and the coefficient of friction between the motorcycle and the track are unchanged.
2. Find the value of \(\tan \alpha\).

4. A light elastic string \(A B\) has natural length 0.8 m and modulus of elasticity 19.6 N . The end \(A\) is attached to a fixed point. A particle of mass 0.5 kg is attached to the end \(B\). The particle is moving with constant angular speed \(\omega\) rad s \(^ { - 1 }\) in a horizontal circle whose centre is vertically below \(A\). The string is inclined at \(60 ^ { \circ }\) to the vertical.

1. Show that the extension of the string is 0.4 m .
2. Find the value of \(\omega\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(6 m g\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant speed \(v\) in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\) and \(O A = 2 a\), as shown in Figure 2 .

1. Show that the extension in the string is \(\frac { 2 } { 5 } a\).
2. Find \(v ^ { 2 }\) in terms of \(a\) and \(g\).

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)

3. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(B\), of mass \(m\), attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\), at a distance \(h\) vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\). The string makes a constant angle \(\theta\) with the downward vertical and \(B\) moves with constant angular speed \(\omega\) about \(O A\).

1. Show that \(\omega ^ { 2 } \leqslant \frac { g } { h }\). The elastic string has natural length \(h\) and modulus of elasticity \(2 m g\).
   Given that \(\tan \theta = \frac { 3 } { 4 }\),
2. find \(\omega\) in terms of \(g\) and \(h\).

1. \begin{figure}[h] \end{figure} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5 l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find

1. the tension in the string,
2. the speed of the ball.

4. \begin{figure}[h] \end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).

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 \(v = \sqrt { } \left( \frac { 1 } { 3 } h g \right)\).

1. \begin{figure}[h] \end{figure} A rough disc is rotating in a horizontal plane with constant angular speed 20 revolutions per minute about a fixed vertical axis through its centre \(O\). A particle \(P\) rests on the disc at a distance 0.4 m from \(O\), as shown in Figure 1. The coefficient of friction between \(P\) and the disc is \(\mu\). The particle \(P\) is on the point of slipping. Find the value of \(\mu\).

3. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string of length 6l. The string passes through a small smooth fixed ring at the point \(A\). The particle \(Q\) is hanging freely at a distance \(l\) vertically below \(A\). The particle \(P\) is moving in a horizontal circle with constant angular speed \(\omega\). The centre \(O\) of the circle is vertically below \(A\). The particle \(Q\) does not move and \(A P\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. Show that

1. \(\theta = 60 ^ { \circ }\)
2. \(\omega = \sqrt { } \left( \frac { 2 g } { 5 l } \right)\)

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 \(O C\).