6.05c Horizontal circles: conical pendulum, banked tracks

4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.

1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
   (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.

7 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-3_599_499_1279_822} A light inextensible string of length \(4 a\) has one end fixed at a point \(P\) and the other end fixed at a point \(Q\), which is vertically below \(P\) and at a distance \(3 a\) from \(P\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. \(R\) moves in a horizontal circle with centre \(Q\) and with the string taut (see diagram).

1. Show that \(Q R = \frac { 7 } { 8 } a\).
2. Find the speed of \(R\) in terms of \(a\) and \(g\).

9 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-4_506_730_625_712} Particles \(P\) and \(Q\), of masses 1.2 kg and 1.5 kg respectively, are attached to the ends of a light inextensible string. The string passes through a small smooth ring which is attached to the ceiling but which is free to rotate. \(P\) rotates at \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle of radius 0.12 m , and \(Q\) hangs vertically in equilibrium (see diagram). Determine

1. the vertical distance below the ring at which \(P\) rotates,
2. the value of \(\omega\).

9 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_671_817_255_623} A particle \(P\) 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .

8 \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_670_819_548_621} A particle \(P\) 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

9 \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-4_433_428_1219_863} A light inextensible string of length 1.4 m has its ends attached to two points \(A\) and \(C\), where \(A\) is 1 m vertically above \(C\). A smooth bead \(B\) of mass 0.2 kg is threaded on the string and rotates in a horizontal circle with the string taut. The distance \(B A\) is 0.8 m (see diagram). Find

1. the tension in the string,
2. the time taken for the bead to perform one complete circle.

8 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-4_604_734_1512_667} A particle \(P\) 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

8 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-4_614_741_1548_662} A particle \(P\) 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).

1. Find the tension in the string in terms of \(m , l\) and \(\omega\).
2. Show that \(\omega ^ { 2 } h = g\).
3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos \theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt{\left(\frac{1}{2}ga\right)}\) from the lowest point of the inside of a fixed hollow smooth sphere of internal radius \(a\) and centre \(O\). The angle between \(OP\) and the downward vertical at \(O\) is denoted by \(\theta\). Show that, as long as \(P\) remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is \(\frac{5}{2}mg(1 + 2\cos\theta)\). [4] Find the speed of \(P\)

1. when it loses contact with the sphere, [3]
2. when, in the subsequent motion, it passes through the horizontal plane containing \(O\). (You may assume that this happens before \(P\) comes into contact with the sphere again.) [3]

\includegraphics{figure_5} A particle 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 \(O\). The point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{(ag)}\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{5}{4}a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt{(ag(1 + 2\cos\alpha))}\). [2]

The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(CBO = 150°\), the tension in the string is the same as it was when the particle was at the point \(A\).

1. Find the value of \(\cos\alpha\). [10]

\includegraphics{figure_5} A particle 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 \(O\). The point \(A\) is such that \(OA = a\) and \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{(ag)}\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{3}{4}a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).

1. Show that, when the string first makes contact with the peg, the speed of the particle is \(\sqrt{(ag(1 + 2\cos\alpha))}\). [2]
2. The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(CBO = 150°\), the tension in the string is the same as it was when the particle was at the point \(A\). Find the value of \(\cos\alpha\). [10]

\includegraphics{figure_3} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T\) N. The acceleration of the particle has magnitude \(7.5 \text{ m s}^{-2}\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\). [4]
2. Find the speed of the particle. [2]

\includegraphics{figure_3} One end of a light inextensible string is attached to a fixed point \(A\) and the other end of the string is attached to a particle \(P\). The particle \(P\) moves with constant angular speed \(5\) rad s\(^{-1}\) in a horizontal circle which has its centre \(O\) vertically below \(A\). The string makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is three times the weight of \(P\).

1. Show that the length of the string is \(1.2\) m. [3]
2. Find the speed of \(P\). [4]

\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4 \text{ kg}\). One end of the string is attached to a fixed point \(A\) \(0.4 \text{ m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3 \text{ m}\) above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3 \text{ m}\) (see diagram).

1. Given that the tension in the string is \(2 \text{ N}\), calculate
   1. the angular speed of the bead, [3]
   2. the magnitude of the contact force exerted on the bead by the surface. [2]
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]

\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4\) kg. One end of the string is attached to a fixed point \(A\) \(0.4\) m above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3\) m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3\) m (see diagram).

1. Given that the tension in the string is \(2\) N, calculate
   1. the angular speed of the bead, [3]
   2. the magnitude of the contact force exerted on the bead by the surface. [2]
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]

A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(\theta\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(θ°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(θ\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

\includegraphics{figure_6} A particle \(P\) of mass \(0.2 \text{ kg}\) is attached to one end of a light inextensible string of length \(0.6 \text{ m}\). The other end of the string is attached to a particle \(Q\) of mass \(0.3 \text{ kg}\). The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length \(0.3 \text{ m}\) and modulus of elasticity \(15 \text{ N}\) joins \(Q\) to a fixed point \(A\) which is \(0.4 \text{ m}\) vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).

1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended. [4]
2. Find the distance \(HP\) given that the angular speed of \(P\) is \(8 \text{ rad s}^{-1}\). [5]

\includegraphics{figure_6} A particle \(P\) of mass 0.2 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 \(A\). The particle \(P\) is also attached to one end of a second light inextensible string of length 0.6 m, the other end of which is attached to a fixed point \(B\) vertically below \(A\). The particle moves in a horizontal circle of radius 0.3 m, which has its centre at the mid-point of \(AB\), with both strings straight (see diagram).

1. Calculate the least possible angular speed of \(P\). [4]
2. Find the greatest possible speed of \(P\). [5]

The string \(AP\) will break if its tension exceeds 8 N. The string \(BP\) will break if its tension exceeds 5 N.

\includegraphics{figure_5} Two particles \(P\) and \(Q\) have masses \(0.4 \text{ kg}\) and \(m \text{ kg}\) respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string of length \(0.5 \text{ m}\) which is inclined at an angle of \(60°\) to the vertical. \(P\) and \(Q\) are joined to each other by a light inextensible vertical string. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string. The string \(BQ\) is taut and horizontal. The particles rotate in horizontal circles about an axis through \(A\) and \(B\) with constant angular speed \(\omega \text{ rad s}^{-1}\) (see diagram). The tension in the string joining \(P\) and \(Q\) is \(1.5 \text{ N}\).

1. Find the tension in the string \(AP\) and the value of \(\omega\). [4]
2. Find \(m\) and the tension in the string \(BQ\). [3]

\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_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).

1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]

A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]

\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]