6.05c Horizontal circles: conical pendulum, banked tracks

2. \begin{figure}[h] \end{figure} One end of a string of length \(3 a\) is attached to a point \(A\) and the other end is attached to a point \(B\) on a smooth horizontal table. The point \(B\) is vertically below \(A\) with \(A B = a \sqrt { 3 }\) A small smooth bead, \(P\), of mass \(m\) is threaded on to the string. The bead \(P\) moves on the table in a horizontal circle, with centre \(B\), with constant speed \(U\). Both portions, \(A P\) and \(B P\), of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.

1. Show that \(A P = 2 a\)
2. Find, in terms of \(m , U\) and \(a\), the tension in the string.
3. Show that \(U ^ { 2 } < a g \sqrt { 3 }\)
4. Describe what would happen if \(U ^ { 2 } > a g \sqrt { 3 }\)
5. State briefly how the tension in the string would be affected if the string were not modelled as being light.

2. \begin{figure}[h] \end{figure} A small smooth ring \(P\), of mass \(m\), is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to a fixed point \(B\) which is vertically above \(A\). The ring moves in a horizontal circle with centre \(A\) and radius \(a\), as shown in Figure 2. The ring moves with constant angular speed \(\sqrt { \frac { 2 g } { 3 a } }\) about \(A B\).
The string remains taut throughout the motion.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find, in terms of \(a\) and \(g\), the angular speed of \(P\) at the instant when it loses contact with the table.
3. Explain how you have used the fact that \(P\) is smooth.

1. A cyclist is travelling around a circular track which is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)

The cyclist moves with constant speed in a horizontal circle of radius \(r\).
In an initial model,

• the cyclist and her cycle are modelled as a particle
• the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction \(\mu\), where \(\mu < \frac { 4 } { 3 }\) Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(V\).
  1. Show that \(V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }\)

In a new simplified model,

• the cyclist and her cycle are modelled as a particle
• the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track

Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(U\).

Find \(U\) in terms of \(r\) and \(g\).

Show that \(U < V\).

1. A girl is cycling round a circular track.

The girl and her bicycle have a combined mass of 55 kg .
The coefficient of friction between the track surface and the tyres of the bicycle is \(\mu\).
The track is banked at an angle of \(15 ^ { \circ }\) to the horizontal.
The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m
The minimum speed at which the girl can cycle round this circle without slipping is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model, find the value of \(\mu\).

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 \(g\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point \(A\) that is vertically above the point \(O\) on a smooth horizontal table, such that \(O A = 40 \mathrm {~cm}\). The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre \(O\), as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.

1. Find (i) the tension in the string,
   (ii) the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find the angular speed of \(P\) at the instant \(P\) loses contact with the table.

4. \begin{figure}[h] \end{figure} One end of a light inextensible string of length \(2 l\) is attached to a fixed point \(A\). A small smooth ring \(R\) of mass \(m\) is threaded on the string and the other end of the string is attached to a fixed point \(B\). The point \(B\) is vertically below \(A\), with \(A B = l\). The ring is then made to move with constant speed \(V\) in a horizontal circle with centre \(B\). The string is taut and \(B R\) is horizontal, as shown in Figure 4.

1. Show that \(B R = \frac { 31 } { 4 }\) Given that air resistance is negligible,
2. find, in terms of \(m\) and \(g\), the tension in the string,
3. find \(V\) in terms of \(g\) and \(l\).

4. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded onto a light inextensible string. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to the fixed point \(B\) such that \(B\) is vertically above \(A\) and \(A B = 6 a\) The ring moves with constant angular speed \(\omega\) in a horizontal circle with centre \(A\). The string is taut and \(B R\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. The ring is modelled as a particle.
Given that \(\tan \theta = \frac { 8 } { 15 }\)

1. find, in terms of \(m\) and \(g\), the magnitude of the tension in the string,
2. find \(\omega\) in terms of \(a\) and \(g\)

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 \(\omega\)

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 \(g\).

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 \(\omega\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-4_547_597_251_735} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\) which is 1.8 m above a smooth horizontal table. The particle moves on the table in a circular path at constant speed with the string taut (see diagram). The particle has a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular velocity is \(0.625 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Show that the radius of the circular path is 0.8 m .
2. Find the magnitude of the normal contact force between the particle and the table. The speed is changed to \(v \mathrm {~ms} ^ { - 1 }\). At this speed the particle is just about to lose contact with the table.
3. Find the value of \(v\).

5 A particle, of mass 6 kg , is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of this circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-4_586_490_541_767} The particle moves in a horizontal circle with an angular speed of 40 revolutions per minute.

1. Show that the angular speed of the particle is \(\frac { 4 \pi } { 3 }\) radians per second.
2. Show that the tension in the string is 67.9 N , correct to three significant figures.
3. Find the radius of the horizontal circle.

6 A particle, of mass 4 kg , is attached to one end of a light inextensible string of length 1.2 metres. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle at a constant speed. The angle between the string and the vertical is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-4_529_554_1580_737}

1. Find the radius of the horizontal circle in terms of \(\theta\).
2. The angular speed of the particle is 5 radians per second. Find \(\theta\).

4 Two light inextensible strings each have one end attached to a particle, \(P\), of mass 6 kg . The other ends of the strings are attached to the fixed points \(B\) and \(C\). The point \(C\) is vertically above the point \(B\). The particle moves, at constant speed, in a horizontal circle, with centre 0.6 m below point \(B\), with the strings inclined at \(40 ^ { \circ }\) and \(60 ^ { \circ }\) to the vertical, as shown in the diagram. Both strings are taut. \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-4_761_542_539_751}

1. As the particle moves in the horizontal circle, the tensions in the two strings are equal. Show that the tension in the strings is 46.4 N , correct to three significant figures.
2. Find the speed of the particle.

5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, \(F\), between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5

1. Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
   Note that there are 1600 metres in one mile.
   5
2. Gary assumes that on a wet day typical values for friction, \(F\), are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.

7 A particle of mass 2.5 kilograms is attached to one end of a light, inextensible string of length 75 cm . The other end of this string is attached to a point \(A\). The particle is also attached to one end of an elastic string of natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). The other end of this string is attached to a point \(B\), which is 60 cm vertically below \(A\). The particle is set in motion so that it describes a horizontal circle with centre \(B\). The angular speed of the particle is \(8 \mathrm { rad } \mathrm { s } { } ^ { - 1 }\) Find \(\lambda\), giving your answer in terms of \(g\).

2 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m . \begin{enumerate}[label=(\alph*)] \item Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\). \item Find the value of \(R\). \item Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns . \item

1. Find the speed of \(Q\) after the collision.
2. Hence show that the collision is inelastic. It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274} In a model, the following modelling assumptions are made.
   For a particular portion of banked track, \(r = 24\).
   (b) Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.
   (c) Explain

3 A right circular cone \(C\) of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of \(C\). The other end of the string is attached to a particle \(P\) of mass 2.5 kg . \(P\) moves in a horizontal circle with constant speed and in contact with the smooth curved surface of \(C\). The extension of the string is 1.5 m .

1. Find the tension in the string.
2. Find the speed of \(P\).

4 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda l x - l m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda x \approx m v ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.