6.05c Horizontal circles: conical pendulum, banked tracks

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

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 \(P\).

5 A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).

1. Given that the tension in the string is 5 N , calculate the speed of \(B\).
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\).

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 \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).

1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.

3 \includegraphics[max width=\textwidth, alt={}, center]{2e4e6e32-eafc-4196-aaa8-42909cc2078e-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).

1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, find the speed of \(P\).
2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.

5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).

1. Find the initial acceleration of \(P\).
2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
3. Find the speed of \(P\).
4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{42de91da-d65e-40e7-8de5-f40eda927850-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
   A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
6. Calculate the weight of the prism.
7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the 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 \(P\).

3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \(( \mathrm { k } + 1 ) \mathrm { a }\).

1. Find the value of \(k\).
2. Find the value of \(\cos \theta\). \includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-06_584_695_264_667} The diagram shows the cross-section \(A B C D\) of a uniform solid object which is formed by removing a cone with cross-section \(D C E\) from the top of a larger cone with cross-section \(A B E\). The perpendicular distance between \(A B\) and \(D C\) is \(h\), the diameter \(A B\) is \(6 r\) and the diameter \(D C\) is \(2 r\).
   1. Find an expression, in terms of \(h\), for the distance of the centre of mass of the solid object from \(A B\).
      The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(A B\) and the downward vertical through \(B\) is \(\theta\).
   2. Given that \(h = \frac { 13 } { 4 } r\), find the value of \(\tan \theta\).

6. \begin{figure}[h] \end{figure} A light inextensible string of length \(14 a\) has its ends attached to two fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = 10 a\). A particle of mass \(m\) is attached to the string at the point \(P\), where \(A P = 8 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\) and with both parts of the string taut, as shown in Figure 3.

1. Show that angle \(A P B = 90 ^ { \circ }\).
2. Show that the time for the particle to make one complete revolution is less than $$2 \pi \sqrt { \left( \frac { 32 a } { 5 g } \right) } .$$

7. \begin{figure}[h] \end{figure} A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.

1. Find \(e\) in terms of \(l\). The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
2. Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\). The particle \(P\) passes through the open top of the tube with speed \(u\).
3. Find \(u\) in terms of \(g\) and \(l\).
4. Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).

3. \begin{figure}[h] \end{figure} A light inextensible string has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). An identical string has one end attached to the fixed point \(B\), where \(B\) is vertically below \(A\) and \(A B = 4 a\), and the other end attached to \(P\), as shown in Figure 2. The particle is moving in a horizontal circle with constant angular speed \(\omega\), with both strings taut and inclined at \(30 ^ { \circ }\) to the vertical. The tension in the upper string is twice the tension in the lower string. Find \(\omega\) in terms of \(a\) and \(g\).

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 \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)

3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).

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 with constant angular speed \(\omega\) in a horizontal circle of radius \(4 a\). The centre of the circle is \(C\), where \(C\) lies on \(A B\) and \(A C = 3 a\), as shown in Figure 3. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 4 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \sqrt { \frac { g } { k a } }\), stating the value of \(k\).

4. \begin{figure}[h] \end{figure} A small smooth bead \(P\) is threaded on a light inextensible string of length \(8 a\). 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 the fixed point \(B\), where \(B\) is vertically above \(A\) and \(A B = 4 a\), as shown in Figure 2. The bead moves with constant angular speed, in a horizontal circle, centre \(A\), with \(A P\) horizontal. The bead remains in contact with the table and both parts of the string, \(A P\) and \(B P\), are taut. The time for \(P\) to complete one revolution is \(S\). Show that \(\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }\)

2. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(2 a\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = a\), as shown in Figure 1. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than \(3 m g\) otherwise the string will break. Given that the time taken by the ball to complete one revolution is \(S\), show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant \(k\).

3. \begin{figure}[h] \end{figure} A fairground ride consists of a cabin \(C\) that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point \(A\) at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point \(B\) on the central axis, where \(A B\) is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,

1. find
   1. the tension in the upper arm of the ride,
   2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
2. State how this would affect the sum of the tensions in the two arms, justifying your answer.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(A\) which lies above a smooth horizontal table. The particle \(P\) moves in a horizontal circle on the table with the string taut. The centre of the circle is the point \(O\) on the table, where \(A O\) is vertical and the string makes a constant angle \(\theta ^ { \circ }\) with \(A O\), as shown in Figure 2. Given that \(P\) moves with constant angular speed \(\sqrt { \frac { 2 g } { a } }\), find the range of possible values of \(\theta\)

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 l\). The other end of the string is attached to a fixed point \(A\) above a smooth horizontal floor. The particle moves in a horizontal circle on the floor with the string taut. The centre \(O\) of the circle is vertically below \(A\) with \(O A = l\), as shown in Figure 2 . The particle moves with constant angular speed \(\omega\) and remains in contact with the floor.
Show that $$\omega \leqslant \sqrt { \frac { g } { l } }$$

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(\mathrm { a } \sqrt { 3 }\). The other end of the string is attached to a fixed point A . The particle P is also attached to one end of a second light inextensible string of length a. The other end of this string is attached to a fixed point B , where B is vertically below A , with \(\mathrm { AB } = \mathrm { a }\). The particle \(P\) moves in a horizontal circle with centre 0 , where 0 is vertically below \(B\).
The particle P moves with constant angular speed \(\omega\), with both strings taut, as shown in Figure 5.

1. Show that the upper string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the lower string makes an angle of \(60 ^ { \circ }\) with the downward vertical.
2. Show that the tension in the upper string is \(\frac { 1 } { 2 } m \sqrt { 3 } \left( 2 g - a \omega ^ { 2 } \right)\).
3. Show that \(\frac { 2 g } { 3 a } < \omega ^ { 2 } < \frac { 2 g } { a }\)
   \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHIL NI GLIUM ION OC
   VEYV SIHI NI III HM ION OC

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 \(\sqrt { \frac { 16 g } { 13 r } }\)

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 3 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a fixed vertical pole. Both strings are taut and \(P\) is moving in a horizontal circle with constant angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). String \(A P\) is inclined at \(30 ^ { \circ }\) to the vertical. String \(B P\) has length 0.4 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 2 . Find the tension in

1. \(A P\),
2. \(B P\).