6.05c Horizontal circles: conical pendulum, banked tracks

\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]

A particle \(P\) of mass \(0.2\) 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 moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]

\includegraphics{figure_7} One end of a light elastic string with modulus of elasticity \(15\) N is attached to a fixed point \(A\) which is \(2\) m vertically above a fixed small smooth ring \(R\). The string has natural length \(2\) m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which moves with constant angular speed \(\omega\) rad s\(^{-1}\) in a horizontal circle which has its centre \(0.4\) m vertically below the ring. \(PR\) makes an acute angle \(\theta\) with the vertical (see diagram).

1. Show that the tension in the string is \(\frac{3}{\cos\theta}\) N and hence find the value of \(m\). [4]
2. Show that the value of \(\omega\) does not depend on \(\theta\). [4]

It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).

1. Find this value of \(\theta\). [4]

\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\text{ kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\text{ m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).

1. Calculate the speed of \(B\). [5]

The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\text{ rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).

1. Calculate the tension in the string. [3]

\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\,\text{kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\,\text{m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).

1. Calculate the speed of \(B\). [5]

The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\,\text{rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).

1. Calculate the tension in the string. [3]

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 \(B\). [3]
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\). [4]

A particle \(P\) of mass \(0.1\text{ kg}\) is attached to one end of a light inextensible string of length \(0.5\text{ m}\). The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface \(0.3\text{ m}\) below \(A\). The tension in the string has magnitude \(T\text{ N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R\text{ N}\).

1. Given that the speed of \(P\) is \(1.5\text{ m s}^{-1}\), calculate \(T\) and \(R\). [4]
2. Given instead that \(T = R\), calculate the angular speed of \(P\). [4]

\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]

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]

\includegraphics{figure_3} Particles \(A\) and \(B\), of masses \(3m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac{2}{3}\sqrt{ga}\). The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac{15}{2}mg\).

1. Find \(\cos \theta\). [3]

\includegraphics{figure_5} A light inextensible string \(AB\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(AC = 3a\) and \(DB = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4}m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(DB\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\). [7]

One end of a light elastic string, of natural length \(12a\) and modulus of elasticity \(kmg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac{2}{3}\sqrt{3ag}\) in a horizontal circle with centre at a distance \(12a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).

1. Find, in terms of \(a\), the extension of the string. [5]
2. Find the value of \(k\). [3]

Two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac{1}{3}\). When the turntable is made to rotate with angular speed \(\frac{2}{5}\sqrt{\frac{g}{a}}\), the system is in limiting equilibrium.

1. Find the tension in the string, in terms of \(m\) and \(g\). [4]
2. Find the value of \(k\). [3]

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). A particle \(Q\) of mass \(km\) is attached to the other end of the string. Particle \(P\) lies on a smooth horizontal table. The string passes through a small smooth hole \(H\) in the table and then passes through a small smooth hole \(H\) in the table. Particle \(P\) moves in a horizontal circle on the surface of the table with constant speed \(\sqrt{\frac{1}{3}ga}\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(HQ = \frac{1}{4}a\).

1. Find, in terms of \(a\), the extension in the string. [4]
2. Find the value of \(k\). [2]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\), 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 \(\sqrt{\frac{g}{a}}\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \((k+1)a\).

1. Find the value of \(k\). [4]
2. Find the value of \(\cos\theta\). [2]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\) on a smooth horizontal plane. A particle \(P\) of mass \(m\) is attached to the other end of the string and moves in a horizontal circle with centre \(O\). The speed of \(P\) is \(\sqrt{\frac{1}{4}ga}\). Find the extension of the string. [4]

\includegraphics{figure_6} A particle \(P\) of mass \(0.05\text{kg}\) is attached to one end of a light inextensible string of length \(1\text{m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04\text{kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8\text{m}\) with angular speed \(\omega\text{rads}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4\text{m}\) also with angular speed \(\omega\text{rads}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).

1. Find the tension in the string \(OP\). [3]

A particle of mass \(2\) kg is attached to one end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(100\) N. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is moving in a horizontal circle about \(O\) with the string taut and with constant angular speed \(5\) radians per second. Find the extension of the string. [3]

\includegraphics{figure_6} A particle \(P\) of mass \(0.05 \text{ kg}\) is attached to one end of a light inextensible string of length \(1 \text{ m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04 \text{ kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8 \text{ m}\) with angular speed \(\omega \text{ rad s}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4 \text{ m}\) also with angular speed \(\omega \text{ rad s}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).

1. Find the tension in the string \(OP\). [3]

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 \(O\).

1. \includegraphics{figure_5a} The particle \(P\) moves in a horizontal circle with a constant angular speed \(\omega\) with the string inclined at \(60°\) to the downward vertical through \(O\) (see diagram). Show that \(\omega^2 = \frac{2g}{a}\). [4]
2. The particle now hangs at rest and is then projected horizontally so that it begins to move in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the downward vertical through \(O\), the angular speed of \(P\) is \(\sqrt{\frac{2g}{a}}\). The string first goes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [6]

A particle 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that \(\omega^2 = \frac{g}{\sqrt{l^2 - r^2}}\) [8]

\includegraphics{figure_1} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length \(13l\). The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find

1. the tension in the string, [3]
2. the speed of the ball. [4]