6.05c Horizontal circles: conical pendulum, banked tracks

3. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(4 l\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(A P\) makes an angle of \(30 ^ { \circ }\) with \(A B\), as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed \(\omega\).

1. Find, in terms of \(m , g , l\) and \(\omega\),
   1. the tension in \(A P\),
   2. the tension in \(B P\).
2. Show that \(\omega ^ { 2 } \geqslant \frac { g \sqrt { 3 } } { 3 l }\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(B\) is vertically below \(A\) and \(A B = l\). The particle is moving with constant angular speed \(\omega\) in a horizontal circle. Both strings are taut and inclined at \(30 ^ { \circ }\) to \(A B\), as shown in Figure 3.

1. 1. Show that the tension in \(A P\) is \(\frac { m \sqrt { 3 } } { 6 } \left( 2 g + l \omega ^ { 2 } \right)\)
   2. Find the tension in \(B P\).
2. Show that the time taken by \(P\) to complete one revolution is less than \(\pi \sqrt { \frac { 2 l } { g } }\)

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle with constant angular speed \(\sqrt { 58.8 } \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre \(O\) of the circle is vertically below \(A\) and the string makes a constant angle \(\theta ^ { \circ }\) with the downward vertical, as shown in Figure 2. Given that the tension in the string is 1.2 mg , find

1. the value of \(\theta\)
2. the length of the string.

1. A rough disc is rotating in a horizontal plane with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. A particle \(P\) is placed on the disc at a distance \(r\) from the axis. The coefficient of friction between \(P\) and the disc is \(\mu\).

Given that \(P\) does not slip on the disc, show that $$\omega \leqslant \sqrt { \frac { \mu g } { r } }$$

2. A light elastic string has natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. One end of the string is attached to a fixed point \(O\). A particle of mass 0.5 kg is attached to the other end of the string. The particle is moving with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with the string stretched. The circle has radius 0.9 m and its centre is vertically below \(O\). The string is inclined at \(60 ^ { \circ }\) to the horizontal. Find

1. the value of \(\lambda\),
2. the value of \(\omega\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(3 a\). The other end of the string is attached to a fixed point \(A\) which is a vertical distance \(a\) above a smooth horizontal table. The particle moves on the table in a circle whose centre \(O\) is vertically below \(A\), as shown in Fig. 1. The string is taut and the speed of \(P\) is \(2 \sqrt { } ( a g )\). Find

1. the tension in the string,
2. the normal reaction of the table on \(P\).

1. \begin{figure}[h] \end{figure} A car moves round a bend in a road which is banked at an angle \(\alpha\) to the horizontal, as shown in Fig. 1. The car is modelled as a particle moving in a horizontal circle of radius 100 m . When the car moves at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), there is no sideways frictional force on the car. Find, in degrees to one decimal place, the value of \(\alpha\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-2_582_798_616_671} A particle of mass \(m\) is attached to the end \(B\) of a light inextensible string. The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\) above the vertex \(V\) of a circular cone of semi-vertical angle \(60 ^ { \circ }\). The axis of the cone is vertical. The particle moves with constant speed \(u\) in a horizontal circle on the smooth surface of the cone. The string makes a constant angle of \(30 ^ { \circ }\) with the vertical (see diagram). The tension in the string and the magnitude of the normal force acting on the particle are denoted by \(T\) and \(R\) respectively. Show that $$T = \frac { m } { \sqrt { } 3 } \left( g + \frac { 2 u ^ { 2 } } { a } \right) ,$$ and find a similar expression for \(R\). Deduce that \(u ^ { 2 } \leqslant \frac { 1 } { 2 } g a\).

One end of a light inextensible string of length \(\frac { 3 } { 2 } a\) is attached to a fixed point \(O\) on a horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(m\). The string passes over a small fixed smooth peg \(A\) which is at a distance \(a\) vertically above \(O\). The system is in equilibrium with \(P\) hanging vertically below \(A\) and the string taut. The particle is projected horizontally with speed \(u\) (see diagram). When \(P\) is at the same horizontal level as \(A\), the tension in the string is \(T\). Show that \(T = \frac { 2 m } { a } \left( u ^ { 2 } - a g \right)\). The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is \(5 : 1\).

1. Show that \(u ^ { 2 } = 5 a g\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is next at the same horizontal level as \(A\).

4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\). Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)

5. A cyclist is travelling around a circular track which is banked at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6 . The cyclist moves with constant speed in a horizontal circle of radius 40 m , without the tyres slipping. Find the maximum speed of the cyclist.

7 \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).

1. Calculate the tension in the string.
2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
3. Calculate \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle \(P\) of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Both strings are taut, the tension in \(A P\) is \(S \mathrm {~N}\) and the tension in \(B P\) is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(4 S = 3 T + 9.8\).
2. Find another equation connecting \(S\) and \(T\) and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}

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 \(P\).

2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.

1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the mass of the car is 700 kg , calculate the power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate

4 A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is fixed at a point \(A\) which is 0.6 m above a smooth horizontal table. \(P\) moves on the table in a circular path whose centre \(O\) is vertically below \(A\).

1. Given that the angular speed of \(P\) is \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
   1. the tension in the string,
   2. the normal reaction between the particle and the table.
   3. Find the greatest possible speed of \(P\), given that the particle remains in contact with the table.

3 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(P Q = 0.8 \mathrm {~m}\). A small smooth bead \(B\), of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius \(0.6 \mathrm {~m} . Q B\) rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram).

1. Show that the tension in the string is 0.1225 N .
2. Find \(\omega\).
3. Calculate the kinetic energy of the bead.

5 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-3_657_549_1219_799} A uniform lamina \(A B C D E\) consists of a square and an isosceles triangle. The square has sides of 18 cm and \(B C = C D = 15 \mathrm {~cm}\) (see diagram).

1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
2. The lamina is freely suspended from \(B\). Calculate the angle that \(B D\) makes with the vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_441_1355_265_394} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A light inextensible string of length 1 m passes through a small smooth hole \(A\) in a fixed smooth horizontal plane. One end of the string is attached to a particle \(P\), of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle \(Q\), of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).

6 Two particles \(A\) and \(B\) are connected by a light inextensible string. Particle \(A\) has mass 1.2 kg and moves on a smooth horizontal table in a circular path of radius 0.6 m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) moves in a horizontal circle in such a way that it is always vertically below \(A\). The angle that the portion of the string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos \theta = \frac { 4 } { 5 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-4_519_803_484_632}

1. Find the time taken for the particles to perform a complete revolution.
2. Find the mass of \(B\). \section*{END OF QUESTION PAPER}

5 One end of a light inextensible string of length 3.5 m is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass \(0.45 \mathrm {~kg} . P\) moves with constant speed in a circular path on the plane with the string taut. The string will break if the tension in it exceeds 70 N . Determine the minimum possible time in which \(P\) can describe a complete circle about \(O\).

7 Two identical light, inextensible strings \(S _ { 1 }\) and \(S _ { 2 }\) are each of length 5 m . Two identical particles \(P\) and \(Q\) are each of mass 1.5 kg . One end of \(S _ { 1 }\) is attached to \(P\). The other end of \(S _ { 1 }\) is attached to a fixed point \(A\) on a smooth horizontal plane. \(P\) moves with constant speed in a horizontal circular path with \(A\) as its centre (see Fig. 1). One end of \(S _ { 2 }\) is attached to \(Q\). The other end of \(S _ { 2 }\) is attached to a fixed point \(B\). \(Q\) moves with constant speed in a horizontal circular path around a point \(O\) which is vertically below \(B\). At any instant, \(B Q\) makes an angle of \(\theta\) with the downward vertical through \(B\) (see Fig. 2). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Given that the angular speed of \(P\) is the same as the angular speed of \(Q\), show that the tensions in \(S _ { 1 }\) and \(S _ { 2 }\) have the same magnitude.
2. You are given instead that the kinetic energy of \(P\) is 39.2 J and that this is the same as the kinetic energy of \(Q\). Determine the difference between the times taken by \(P\) and \(Q\) to complete one revolution. Give your answer in an exact form.

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 1.8 m . The other end of the string is attached to a fixed point, \(O\), on a smooth horizontal plane. Initially, \(P\) is moving with a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with \(O\) as its centre.

1. 1. Find the magnitude of the acceleration of \(P\).
   2. State the direction of the acceleration of \(P\). A force is now applied to \(P\) in such a way that its angular velocity increases. At the instant that the angular velocity reaches \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\), the string breaks.
2. 1. Find the speed with which \(P\) is moving at the instant that the string breaks.
   2. Find the tension in the string at the instant that the string breaks. After the string has broken \(P\) starts to move directly up a smooth slope which is fixed to the plane and inclined at an angle \(\theta ^ { \circ }\) above the horizontal. Particle \(P\) moves a distance of 20 m up the slope before coming to instantaneous rest.
3. Use an energy method to determine the value of \(\theta\).

7 It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~ms} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-5_414_624_356_242} In a model, the following modelling assumptions are made.

• The track is smooth.
• The car is a particle.
• The car follows a horizontal circular path with radius \(r \mathrm {~m}\).
  1. Show that, according to the model, \(\sqrt { 3 } \mathrm { v } ^ { 2 } = \mathrm { gr }\).

For a particular portion of banked track, \(r = 24\).

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.

Explain

4 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 \(P\).

2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-2_344_340_1418_842}

1. Show that the tension in the string is 22.6 N , correct to three significant figures.
2. Find the speed of the particle.