6.05c Horizontal circles: conical pendulum, banked tracks

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 0.5 kg is attached to one end of a light inextensible string of length 1.5 m. The other end of the string is attached to a fixed point \(A\). The particle is moving, with the string taut, in a horizontal circle with centre \(O\) vertically below \(A\). The particle is moving with constant angular speed 2.7 rad s\(^{-1}\). Find

1. the tension in the string, [4]
2. the angle, to the nearest degree, that \(AP\) makes with the downward vertical. [3]

\includegraphics{figure_3} A small ball \(P\) of mass \(m\) is attached to the ends of two light inextensible strings of length \(l\). The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(AP\) is perpendicular to \(BP\) as shown in Figure 3. The system rotates about the line \(AB\) with constant angular speed \(\omega\). The ball moves in a horizontal circle.

1. Find, in terms of \(m\), \(g\), \(l\) and \(\omega\), the tension in \(AP\) and the tension in \(BP\). [8]
2. Show that \(\omega^2 \geq \frac{g\sqrt{2}}{l}\). [2]

\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to two light inextensible strings. The other ends of the string are attached to fixed points \(A\) and \(B\). The point \(A\) is a distance \(h\) vertically above \(B\). The system rotates about the line \(AB\) with constant angular speed \(\omega\). Both strings are taut and inclined at \(60°\) to \(AB\), as shown in Fig. 4. The particle moves in a circle of radius \(r\).

1. Show that \(r = \frac{\sqrt{3}}{2}h\). [2]
2. Find, in terms of \(m\), \(g\), \(h\) and \(\omega\), the tension in \(AP\) and the tension in \(BP\). [8]

The time taken for \(P\) to complete one circle is \(T\).

1. Show that \(T < \pi\sqrt{\left(\frac{2h}{g}\right)}\). [4]

\includegraphics{figure_1} A metal ball \(B\) 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 ball \(B\) moves in a horizontal circle with centre \(O\) vertically below \(A\), as shown in Fig. 1. The string makes a constant angle \(\alpha°\) with the downward vertical and \(B\) moves with constant angular speed \(\sqrt{(2gk)}\), where \(k\) is a constant. The tension in the string is \(3mg\). By modelling \(B\) as a particle, find

1. the value of \(\alpha\), [4]
2. the length of the string. [5]

A car moves round a bend which is banked at a constant angle of \(10°\) to the horizontal. When the car is travelling at a constant speed of \(18 \text{ m s}^{-1}\), there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius \(r\) metres. Calculate the value of \(r\). [6]

\includegraphics{figure_1} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings \(AP\) and \(BP\) each of length \(l\). The ends \(A\) and \(B\) are attached to fixed points, with \(A\) vertically above \(B\) and \(AB = \frac{3}{4}l\), as shown in Fig. 1. The particle \(P\) moves in a horizontal circle with constant angular speed \(\omega\). The centre of the circle is the mid-point of \(AB\) and both strings remain taut.

1. Show that the tension \(AP\) is \(\frac{1}{6}m(3l\omega^2 + 4g)\). [7]
2. Find, in terms of \(m\), \(l\), \(\omega\) and \(g\), an expression for the tension in \(BP\). [2]
3. Deduce that \(\omega^2 \geq \frac{4g}{3l}\). [2]

\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]

A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$gr^2 = v^2\sqrt{l^2 - r^2}.$$ [9]

\includegraphics{figure_2} A particle \(P\) of mass \(m\) moves on the smooth inner surface of a hemispherical bowl of radius \(r\). The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed \(\sqrt{\left(\frac{3g}{2r}\right)}\) in a horizontal circle at depth \(d\) below the centre of the bowl.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the bowl on \(P\). [4]
2. Find \(d\) in terms of \(r\). [4]

\includegraphics{figure_1} A particle \(Q\) of mass 5 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a vertical pole. Each string has length 0.6 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 1. Both strings are taut and \(Q\) is moving in a horizontal circle with constant angular speed 10 rad s\(^{-1}\). Find the tension in

1. \(AQ\),
2. \(BQ\). [10]

\includegraphics{figure_1} A cone of semi-vertical angle \(60°\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60°\) with the horizontal, as shown in Figure 1.

1. Find the tension in the string, in terms of \(m\), \(l\), \(\omega\) and \(g\). [7]

The particle remains on the surface of the cone.

1. Show that the time for the particle to make one complete revolution is greater than $$2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$$ [6]

A particle, of mass 0.8 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 \(35°\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics{figure_4} The particle moves in a horizontal circle and completes 20 revolutions each minute.

1. Find the angular speed of the particle in radians per second. [2 marks]
2. Find the tension in the string. [3 marks]
3. Find the radius of the horizontal circle. [4 marks]

A particle \(P\), of mass \(6\) kg, is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass \(8\) kg, is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring, and the particle \(P\) moves with speed \(5 \text{ m s}^{-1}\) in a horizontal circle, as shown in the diagram. The angle between \(OP\) and the vertical is \(\theta\). \includegraphics{figure_4}

1. Find the tension in the string. [1 mark]
2. Find \(\theta\). [3 marks]
3. Find the radius of the horizontal circle. [4 marks]

\includegraphics{figure_3} 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 \(PQ = 0.8\) 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 m. \(QB\) rotates with constant angular speed \(\omega\) rad s\(^{-1}\) (see diagram).

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

\includegraphics{figure_8} A conical shell has radius 6 m and height 8 m. The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg, is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).

1. The frictional force on the particle is \(F\) N, and the normal force of the shell on the particle is \(R\) N. It is given that the speed of the particle is 4.5 ms\(^{-1}\), which is the smallest possible speed for the particle not to slip.
   1. By resolving vertically, show that \(4F + 3R = 19.6\). [2]
   2. By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures. [6]
2. Find the largest possible angular speed of the shell for which the particle does not slip. [6]

One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semi-vertical angle \(45°\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(\omega\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).

1. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac{1}{2}ml(\sqrt{2}g + l\omega^2)\). [6]
2. When the string has length 0.8 m, calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone. [4]

A smooth solid cone of semi-vertical angle \(60°\) is fixed to the ground with its axis vertical. 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 fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30°\) (see diagram).

1. Show that the magnitude of the contact force between the cone and the particle is \(\frac{1}{4}m(2\sqrt{3}g - 3a\omega^2)\). [6]
2. Given that \(a = 0.5\) m and \(m = 3.5\) kg, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string. [3]

A bird of mass 0.5 kg, flying around a vertical feeding post at a constant speed of 6 ms\(^{-1}\), banks its wings to move in a horizontal circle of radius 2 m. The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown. \includegraphics{figure_1} Modelling the bird as a particle, find, to the nearest degree, the angle that its wings make with the vertical. [7 marks]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\omega\) rad s\(^{-1}\). \includegraphics{figure_2} If the angle \(\theta\) which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of \(\omega\). [7 marks]

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]

Aliya, whose mass is \(m\) kg, is playing rounders. She rounds the first base at a speed of \(v\) ms\(^{-1}\), making the turn on a horizontal circular path of radius \(r\) m.

1. Write down, in terms of \(m\), \(v\) and \(r\), the magnitude of the horizontal force acting on her. [1 mark]
2. Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle \(\theta\) to the vertical, where \(\tan \theta = \frac{v^2}{gr}\). [6 marks]

A light inelastic string of length \(l\) m passes through a small smooth ring which is fixed at a point \(O\) and is free to rotate about a vertical axis through \(O\). Particles \(P\) and \(Q\), of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.

1. \(Q\) describes a horizontal circle with centre \(P\), while \(P\) hangs at rest at a depth \(d\) m below \(O\). Show that \(d = \frac{2l}{5}\). [6 marks]
2. \(P\) and \(Q\) now both move in horizontal circles with the same angular velocity \(\omega\) rad s\(^{-1}\) about a vertical axis through \(O\). Show that \(OQ = \frac{3l}{5}\) m. [7 marks]

\includegraphics{figure_5}

A particle \(P\) of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\pi\) rad s\(^{-1}\). The natural length of the string is 1 m and, while \(P\) is in motion, the distance \(OP = 1.15\) m.

1. Calculate, to 3 significant figures, the modulus of elasticity of the string. [6 marks]

The motion now ceases and \(P\) hangs at rest vertically below \(O\).

1. Show that the extension in the string in this position is about 13 cm. [3 marks]

A car of mass \(m\) kg moves round a curve of radius \(r\) m on a road which is banked at an angle \(\theta\) to the horizontal. When the speed of the car is \(u\) ms\(^{-1}\), the car experiences no sideways frictional force. Given that \(\tan \theta = \frac{u^2}{gr}\), show that the sideways frictional force on the car when its speed is \(\frac{u}{2}\) ms\(^{-1}\) has magnitude \(\frac{3}{4}mg \sin \theta\) N. [10 marks]