Circular Motion 1

1. Find the value of \(\beta\). [2]

1. Find the value of \(\beta\). [2]

5 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-3_365_679_264_733} A uniform metal frame \(O A B C\) is made from a semicircular \(\operatorname { arc } A B C\) of radius 1.8 m , and a straight \(\operatorname { rod } A O C\) with \(A O = O C = 1.8 \mathrm {~m}\) (see diagram).

1. Calculate the distance of the centre of mass of the frame from \(O\). A uniform semicircular lamina of radius 1.8 m has weight 27.5 N . A non-uniform object is formed by attaching the frame \(O A B C\) around the perimeter of the lamina. The object is freely suspended from a fixed point at \(A\) and hangs in equilibrium. The diameter \(A O C\) of the object makes an angle of \(22 ^ { \circ }\) with the vertical.
2. Calculate the weight of the frame.

1. It is given that when the ball moves with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the tension in the string \(Q B\) is three times the tension in the string \(P B\). Calculate the radius of the circle. The ball now moves along this circular path with the minimum possible speed.
2. State the tension in the string \(P B\) in this case, and find the speed of the ball.

2
The ends of two light inextensible strings of length 0.7 m are attached to a particle \(P\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\). The particle \(P\) moves in a horizontal circle which has its centre at the mid-point of \(A B\). Both strings are inclined at \(60 ^ { \circ }\) to the vertical. The tension in the string attached to \(A\) is 6 N and the tension in the string attached to \(B\) is 4 N (see diagram).

1. Find the mass of \(P\).
2. Calculate the speed of \(P\).

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 \(k\).

1. Find, in terms of \(a\), the distance of \(B\) below the ring. [3]

1. Find, in terms of \(a\), the distance of \(B\) below the ring. [3]

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 \(\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }\)

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 \(B\).

6 A particle \(P\) of mass 0.6 kg is released from rest at a point above ground level and falls vertically. The motion of \(P\) is opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\). Immediately before \(P\) reaches the ground, \(v = 1.95\).

1. Calculate the time after its release when \(P\) reaches the ground. \(P\) is now projected horizontally with speed \(1.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across a smooth horizontal surface. The motion of \(P\) is again opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\).
2. Calculate the distance \(P\) travels after projection before coming to rest.

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 horizontal circular disc rotates with constant angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about its centre \(O\). A particle of mass 0.05 kg is placed on the disc at a distance 0.4 m from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc.

1 A particle \(P\) of mass 0.6 kg is on the rough surface of a horizontal disc with centre \(O\). The distance \(O P\) is 0.4 m . The disc and \(P\) rotate with angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis which passes through \(O\). Find the magnitude of the frictional force which the disc exerts on the particle, and state the direction of this force.

11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.

1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
3. 1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
   2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?

A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{8}{3}ga}\). Find \(x\) in terms of \(a\). [6]

4
A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).

1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
2. Find the speed of the particle.
3. Find the time taken for the particle to complete one revolution.

\includegraphics{figure_12} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.

1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]

\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]

4 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_576_826_258_662} A hollow cone with semi-vertical angle \(45 ^ { \circ }\) is fixed with its axis vertical and its vertex \(O\) downwards. A particle \(P\) of mass 0.3 kg moves in a horizontal circle on the inner surface of the cone, which is smooth. \(P\) is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is attached to the cone at \(O\) (see diagram). The string is taut and rotates at a constant angular speed of \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of \(P\).
2. Find the tension in the string and the force exerted on \(P\) by the cone.

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 \(g\).

1 A propeller shaft has constant angular acceleration. It turns through 160 radians as its angular speed increases from \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find

1. the angular acceleration of the propeller shaft,
2. the time taken for this increase in angular speed.

1 A wheel rotating about a fixed axis is slowing down with constant angular deceleration. Initially the angular speed is \(24 \mathrm { rad } \mathrm { s } ^ { - 1 }\). In the first 5 seconds the wheel turns through 96 radians.

1. Find the angular deceleration.
2. Find the total angle the wheel turns through before coming to rest.

1 A uniform square lamina, of mass 4.5 kg and side 0.6 m , is rotating about a fixed vertical axis which is perpendicular to the lamina and passes through its centre. A stationary particle becomes attached to the lamina at one of its corners, and this causes the angular speed of the lamina to change instantaneously from \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle. The lamina then slows down with constant angular deceleration. It turns through 36 radians as its angular speed reduces from \(1.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to zero.
2. Find the time taken for the lamina to come to rest.

1 A particle is moving in a circle of radius 2 m . At time \(t \mathrm {~s}\) its velocity is \(\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the magnitude of the resultant acceleration of the particle when \(t = 4\).

A roundabout in a playground can be modeled as a horizontal circular platform with centre \(O\). The roundabout is free to rotate about a vertical axis through \(O\). A child sits without slipping on the roundabout at a horizontal distance of 1.5 m from \(O\) and completes one revolution in 2.4 seconds.

1. Calculate the speed of the child. [3]
2. Find the magnitude and direction of the acceleration of the child. [3]

\includegraphics{figure_4} Two coplanar discs, of radii \(0.5\) m and \(0.3\) m, rotate about their centres \(A\) and \(B\) respectively, where \(AB = 0.8\) m. At time \(t\) seconds the angular speed of the larger disc is \(\frac{1}{2}t\) rad s\(^{-1}\) (see diagram). There is no slipping at the point of contact. For the instant when \(t = 2\), find

1. the angular speed of the smaller disc, [2]
2. the magnitude of the acceleration of a point \(P\) on the circumference of the larger disc, and the angle between the direction of this acceleration and \(PA\). [7]

1. Show that \(v ^ { 2 } = u ^ { 2 } - 4 a g\).
2. The ratio of the tensions in the string when the bead is at the two points \(A\) and \(B\) is \(2 : 5\).
   1. Find \(u\) in terms of \(g\) and \(a\).
   2. Find the ratio \(u : v\).

3 A light inextensible string has length \(2 a\). One end of the string is attached to a fixed point \(O\) and a particle of mass \(m\) is attached to the other end. Initially, the particle is held at the point \(A\) with the string taut and horizontal. The particle is then released from rest and moves in a circular path. Subsequently, it passes through the point \(B\), which is directly below \(O\). The points \(O , A\) and \(B\) are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-3_426_437_575_772}

1. Show that the speed of the particle at \(B\) is \(2 \sqrt { a g }\).
2. Find the tension in the string as the particle passes through \(B\). Give your answer in terms of \(m\) and \(g\).

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 \(A\).

A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac{1}{2}u\) the angle \(\theta\) between \(OP\) and the downward vertical satisfies the equation $$8ga(1 - \cos\theta) = 3u^2.$$ [2] Find, in terms of \(m\), \(u\), \(a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position. [4]

4 The diagram shows two particles P and Q , of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m . Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to \(\theta ^ { \circ }\). The pulley, P and Q all lie in the same vertical plane. \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-4_499_492_559_251} Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P .

1. Given that P is in equilibrium at the point where \(\theta = \alpha\), determine the value of \(\alpha\). Particle P is now released from rest at the point on the surface where \(\theta = 35\), and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.
2. By considering energy, determine the speed of P at the instant when \(\theta = 45\).
3. State one modelling assumption you have made in determining the answer to part (b).

The diagram shows two identical particles, each of mass \(m\) kg, connected by a thin, light inextensible string. \(P\) slides on the surface of a smooth right circular cylinder fixed with its axis, through \(O\), horizontal. \(Q\) moves vertically. \(OP\) makes an angle \(\theta\) radians with the horizontal. \includegraphics{figure_6} The system is released from rest in the position where \(\theta = 0\).

1. Show that the vertical distance moved by \(Q\) is \(\frac{\theta}{\sin \theta}\) times the vertical distance moved by \(P\). [4 marks]
2. In the position where \(\theta = \frac{\pi}{6}\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left(1-\frac{\pi}{6}\right)mg\) N. [9 marks]

A cyclist is travelling around a circular track which is banked at 25° 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. [10]

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.

A cyclist is travelling around a circular track which is banked at 25° 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. [10]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(mg\) N. The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60°\) with the vertical,

1. show that \(OP = 3l\) m. [4 marks]
2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\). [4 marks]

4 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_337_526_262_726} \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_111_116_486_1308} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The particle \(P\) moves in a horizontal circle which has centre \(O\). It is given that \(A O\) is vertical and that angle \(O A P\) is \(60 ^ { \circ }\) (see diagram). Calculate the speed of \(P\). [6]

In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)

1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
2. Use your calculator to find \(\alpha\) [1 mark]
3. Find the value of \(v\) [4 marks]

A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l\) m. The speed of \(P\) is 4 m s\(^{-1}\), and the radius of the circle in which it moves is 2l m. Calculate \(l\). [4]

1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).

1. Find the radius of this circle.
2. Find the modulus of elasticity of the string.

8 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 \(v \mathrm {~ms} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda | x - | m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda \mathrm { x } \approx \mathrm { mv } ^ { 2 }\).
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 \(7 \mathrm {~ms} ^ { - 1 }\). 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.

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

The gravitational attraction \(F\) N between two point masses \(m_1\) kg and \(m_2\) kg at a distance \(x\) m apart is given by \(F = \frac{km_1m_2}{x^2}\), where \(k\) is a constant. Given that a small body of mass \(1\) kg experiences a force of \(g\) N at the surface of the Earth, which has radius \(R\) m and mass \(M\) kg,

1. show that \(k = \frac{gR^2}{M}\). [2 marks]

A small communications satellite of mass \(m\) kg is put into a circular orbit of radius \(r\) m around the Earth. Modelling the Earth as a particle of mass \(M\) kg, and using the value of \(k\) from (a),

1. prove that the period of rotation, \(T\) s, of the satellite is given by \(T = \frac{2\pi}{R}\sqrt{\frac{r^3}{g}}\). [4 marks]

To cover transmission to any point on the Earth, three small satellites \(X\), \(Y\) and \(Z\), each of mass \(m\) kg, are placed in a common circular orbit of radius \(r\) and form an equilateral triangle as shown. \includegraphics{figure_6}

1. Show on a copy of the diagram the direction of the three forces acting on \(X\). [1 mark]
2. State, with a reason, the direction of the resultant force on \(X\). [2 marks]
3. Show that the period of rotation of \(X\) is given by \(T\sqrt{\frac{3M}{2M + m\sqrt{3}}}\) s, where \(T\) s is the period found in (b). [7 marks]

1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20 N . Find how many revolutions the particle makes per minute.

1 A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(O\) of a smooth horizontal plane. \(P\) moves on the plane at constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circle with centre \(O\). Calculate the tension in the string.

\includegraphics{figure_3} One end of a light inextensible string of length \(0.2 \text{ m}\) is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass \(0.6 \text{ kg}\) is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \text{ m s}^{-1}\), with the string taut and making an angle of \(30°\) to the horizontal (see diagram).

1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\). [4]
2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\). [3]

\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]

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

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 \(\omega\).

2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)

2 A particle has an angular speed of 72 revolutions per minute.
Find the angular speed in radians per second.
Circle your answer.
[0pt] [1 mark] \(\frac { 6 \pi } { 5 } \quad \frac { 12 \pi } { 5 } \quad 12 \pi \quad 24 \pi\)

4
2.4
&

B1 for each of two correct statements about the models.
If commenting on the accuracy of (a), must emphasise that (a) is very inaccurate or at least quite inaccurate
Do not allow e.g.
- model (a) is not very effective
- Neither model is accurate
- (a) and (b) are not very accurate
Clear comparison between the accuracy of the two models (must emphasise that (b) is fairly accurate or considerably more accurate than (a)), or other suitable distinct second comment
Do not allow e.g.
- model (b) is more accurate than model (a)
- (b) is not accurate
Do not allow statement claiming that resistance is proportional to speed, or to speed \({ } ^ { 2 }\)
	Suitable comments for (a): - is very inaccurate - predicted speed is nearly three times the actual value - constant resistance is not a suitable model - both models underestimate the resistance (as top speed is lower than expected) For the linear model (b) - is fairly accurate (but probably underestimates the resistance at higher speeds) - resistance is not proportional to speed but is a much better model than constant resistance

1 A railway engine of mass 50000 kg travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal circular track of radius 1250 m . Find the magnitude of the horizontal force on the engine.

\includegraphics{figure_7} A light inextensible string of length 8 m is threaded through a smooth fixed ring, \(R\), and carries a particle at each end. One particle, \(P\), of mass 0.5 kg is at rest at a distance 3 m below \(R\). The other particle, \(Q\), is rotating in a horizontal circle whose centre coincides with the position of \(P\) (see diagram). Find the angular speed and the mass of \(Q\). [8]

1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
2. Show that
   1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
   2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
4. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.

1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
2. Calculate the tension in the string.
3. Calculate the speed of \(P\).

A motorcyclist rides in a cylindrical well of radius 5 m. He maintains a horizontal circular path at a constant speed of 10 ms\(^{-1}\). The coefficient of friction between the wall and the wheels of the cycle is \(\mu\). \includegraphics{figure_1} Modelling the cyclist and his machine as a particle in contact with the wall, show that he will not slip downwards provided that \(\mu \geq 0.49\). [7 marks]

\(ABC\) is a uniform semicircular arc with diameter \(AC = 0.5\) m. The arc rotates about a fixed axis through \(A\) and \(C\) with angular speed \(2.4\) rad s\(^{-1}\). Calculate the speed of the centre of mass of the arc. [3]

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 \(\alpha\).

A uniform rod \(PQ\), of mass \(m\) and length \(2a\), is made to rotate in a vertical plane with constant angular speed \(\sqrt{\frac{g}{a}}\) about a fixed smooth horizontal axis through the end \(P\) of the rod. Show that, when the rod is inclined at an angle \(\theta\) to the downward vertical, the magnitude of the force exerted on the axis by the rod is \(2mg|\cos(\frac{1}{2}\theta)|\). [8]

\includegraphics{figure_6} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg, by three light rods where the length of rod \(AP\) is 1.5 m and the length of rod \(PQ\) is 0.75 m. Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A\), \(B\), \(P\) and \(Q\) are coplanar. The rod \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).

1. Find the tension in the rod \(PQ\). [2]
2. Find \(\omega\). [3]
3. Find the speed of \(P\). [1]
4. Find the tension in the rod \(AP\). [3]
5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]