6.05c Horizontal circles: conical pendulum, banked tracks

A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.

1. Show that when its speed is 8.28 ms\(^{-1}\), there is no sideways force acting on the car. [4 marks]
2. When the speed of the car is 12.5 ms\(^{-1}\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]

A cyclist travels on a banked track inclined at \(8°\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v\) ms\(^{-1}\). If there is no sideways frictional force on the cycle, calculate the value of \(v\). [6 marks]

A particle \(P\), of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is 0·5. \(P\) is connected to a particle \(Q\), of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point \(O\) on the table, such that the distance \(OQ\) is 0·4 m. \(Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium. \includegraphics{figure_5}

1. Calculate the angle \(\theta\) which \(OQ\) makes with the vertical. [4 marks]
2. Show that the speed of \(Q\) is 1·33 ms\(^{-1}\). [3 marks]

The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed 0·84 ms\(^{-1}\), at a constant distance \(r\) m from \(O\) but tending to slip away from \(O\).

1. Find the value of \(r\). [5 marks]

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]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}

1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
2. Find \(\omega\) [3 marks]
3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]

A cyclist in a road race is travelling around a bend on a horizontal circular path of radius 15 metres and is prevented from skidding by a frictional force. The frictional force has a maximum value of 500 newtons. The total mass of the cyclist and his cycle is 75 kg Assume that the cyclist travels at a constant speed.

1. Work out the greatest speed, in km h\(^{-1}\), at which the cyclist can travel around the bend. [4 marks]
2. With reference to the surface of the road, describe one limitation of the model. [1 mark]

A small coin is placed at a point \(C\) on a rough horizontal turntable, with centre \(O\), as shown in the diagram below. \includegraphics{figure_9} The mass of the coin is 3.6 grams. The distance \(OC\) is 20 cm The turntable rotates about a vertical axis through \(O\), with constant angular speed \(\omega\) radians per second.

1. Draw a diagram to show all the forces acting on the coin. [1 mark]
2. The maximum value of friction is 0.01 newtons and the coin does not slip during the motion. Find the maximum value of \(\omega\) Give your answer to two significant figures. [4 marks]
3. State one modelling assumption you have made to answer part (b). [1 mark]

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 conical pendulum consists of a light string and a particle of mass \(m\) kg The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. \includegraphics{figure_3} Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(T \cos \theta = mr\omega^2\) \quad \(\square\) \(T \sin \theta = mr\omega^2\) \quad \(\square\) \(T \cos \theta = \frac{m\omega^2}{r}\) \quad \(\square\) \(T \sin \theta = \frac{m\omega^2}{r}\) \quad \(\square\)

\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]

\includegraphics{figure_12} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P. When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.

1. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m(r\omega^2\cos\theta - g\sin\theta)\). [6]
2. Find a similar expression, in terms of \(r\), \(\omega\), \(m\), \(g\) and \(\theta\), for the tension in the part of the string AP. [2]

It is given that AB = 5a and AP = 4a.

1. Show that \(16a\omega^2 > 5g\). [3]

The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.

1. Find the period of the motion of the ring, giving your answer in terms of \(a\), \(g\) and \(\pi\). [5]

\includegraphics{figure_4} The diagram shows a particle P, of mass 0.1 kg, which is attached by a light inextensible string of length 0.5 m to a fixed point O. P moves with constant angular speed 5 rad s\(^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical.

1. Determine the tension in the string. [3]
2. Find the value of \(\theta\). [2]
3. Find the kinetic energy of P. [2]

\includegraphics{figure_3} The diagram shows a particle P, of mass 0.2 kg, which is attached by a light inextensible string of length 0.75 m to a fixed point O. Particle P moves with constant angular speed \(\omega \text{ rad s}^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N.

1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree. [2]
2. Determine the greatest possible value of \(\omega\). [3]

\includegraphics{figure_13} A conical shell, of semi-vertical angle \(\alpha\), is fixed with its axis vertical and its vertex V upwards. A light inextensible string passes through a small smooth hole at V and a particle P of mass 4 kg hangs in equilibrium at one end of the string. The other end of the string is attached to a particle Q of mass 25 kg which moves in a horizontal circle at constant angular speed \(2.8 \text{ rad s}^{-1}\) on the smooth outer surface of the shell at a vertical depth \(h\) m below V (see diagram).

1. Show that \(k_1 h \sin^2 \alpha + k_2 \cos^2 \alpha = k_3 \cos \alpha\), where \(k_1\), \(k_2\) and \(k_3\) are integers to be determined. [7]
2. Determine the greatest value of \(h\) for which Q remains in contact with the shell. [3]

\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]

Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}

1. Show that the tension in the string is 6.37 N. [4]
2. Find the speed of R. [4]

A vehicle of mass 1200 kg is moving with a constant speed of \(40\text{ ms}^{-1}\) around a horizontal circular path which is on a test track banked at an angle of 60° to the horizontal. There is no tendency to sideslip at this speed. The vehicle is modelled as a particle.

1. Calculate the normal reaction of the track on the vehicle. [3]
2. Determine
   1. the radius of the circular path,
   2. the angular speed of the vehicle and clearly state its units. [6]
3. What further assumption have you made in your solution to (b)? Briefly explain what effect this assumption has on the radius of the circular path. [2]

The diagram below shows a particle \(P\), of mass 2.5 kg, attached by means of two light inextensible strings fixed at points \(A\) and \(B\). Point \(A\) is vertically above point \(B\). \(BP\) makes an angle of \(60°\) with the upward vertical and \(AP\) is inclined at an angle \(\theta\) to the downward vertical where \(\cos\theta = 0.8\). The particle \(P\) describes a horizontal circle with constant angular speed \(\omega\) radians per second about centre \(C\) with both strings taut. \includegraphics{figure_7} The tension in the string \(BP\) is 39.2 N.

1. Calculate the tension in the string \(AP\). [4]
2. Given that the length of the string \(AP\) is 1.5 m, find the value of \(\omega\). [5]
3. Calculate the kinetic energy of \(P\). [3]

The diagram below shows a hollow cone, of base radius \(5\) m and height \(12\) m, which is fixed with its axis vertical and vertex \(V\) downwards. A particle \(P\), of mass \(M\) kg, moves in a horizontal circle with centre \(C\) on the smooth inner surface of the cone with constant speed \(v = 3\sqrt{g}\) ms\(^{-1}\). \includegraphics{figure_3}

1. Show that the normal reaction of the surface of the cone on the particle is \(\frac{13Mg}{5}\) N. [4]
2. Calculate the length of \(CP\) and hence determine the height of \(C\) above \(V\). [6]

A particle of mass \(m\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set in motion such that it moves in a horizontal circle of radius 2 m with constant speed 4.8 ms\(^{-1}\). Calculate the angle the string makes with the vertical. [6]

\includegraphics{figure_4} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length \(1\) m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length \(1\) m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius \(0.6\) m with constant angular velocity \(\omega\) rad s\(^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.

1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
2. Show that
   1. \(T_1 = \frac{4g}{5}(m_1 + m_2)\). [2]
   2. \(\omega^2 = \frac{4g(m_1 + 2m_2)}{4m_1}\). [3]
3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2]

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.

1. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega\) rad s\(^{-1}\) to zero. [5]

\includegraphics{figure_3} 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 \(g\). [6]

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]
2. Find the speed of P. [7]

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 \(\sqrt{\frac{g}{2}a}\). Find \(x\) in terms of \(a\). [6]

\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]