Vertical circle – string/rod (tension and energy)

6. A particle \(P\) 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 particle is hanging in equilibrium below \(O\) when it receives a horizontal impulse giving it a speed \(u\), where \(u ^ { 2 } = 3 g a\). The string becomes slack when \(P\) is at the point \(B\). The line \(O B\) makes an angle \(\theta\) with the upward vertical.

1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
   (9)
2. Show that the greatest height of \(P\) above \(B\) in the subsequent motion is \(\frac { 4 a } { 27 }\).
   (6)

4 \includegraphics[max width=\textwidth, alt={}, center]{ae8d874a-5c1d-45bb-b853-d12006004b7f-2_519_583_1384_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).

1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
2. Find the speed of the bead as it leaves the wire at \(B\).
3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).

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 \(A\).

2 A small bead \(B\) of mass \(m\) is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius \(a\) and centre \(O\). The highest point of the circle is \(A\). The bead is slightly displaced from rest at \(A\). When angle \(A O B = \theta\), where \(\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\), the force exerted on the bead by the wire has magnitude \(R _ { 1 }\). When angle \(A O B = \pi + \theta\), the force exerted on the bead by the wire has magnitude \(R _ { 2 }\). Show that \(R _ { 2 } - R _ { 1 } = 4 m g\).

3 A rough circular hoop, with centre O and radius 1 m , is fixed in a vertical plane. A , B and C are points on the hoop such that A and C are at the same horizontal level as O , and OB makes an angle of \(25 ^ { \circ }\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-4_650_729_404_251} A bead P of mass 0.3 kg is threaded onto the hoop. P is projected vertically downwards from A on two separate occasions.

• The first time, when P is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at B .
• The second time, when P is projected with a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it first comes to rest at C .

The situation is modelled by assuming that during the motion of P the magnitude of the frictional force exerted by the hoop on P is constant.

1. Determine the value of \(v\).
2. Comment on the validity of the modelling assumption used in this question.

5 A young man of mass 60 kg swings on a trapeze. A simple model of this situation is as follows. The trapeze is a light seat suspended from a fixed point by a light inextensible rope. The man's centre of mass, G , moves on an arc of a circle of radius 9 m with centre O , as shown in Fig. 5. The point C is 9 m vertically below O . B is a point on the arc where angle COB is \(45 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the gravitational potential energy lost by the man if he swings from B to C . In this model it is also assumed that there is no resistance to the man's motion and he starts at rest from B.
2. Using an energy method, find the man's speed at C . A new model is proposed which also takes into account resistance to the man's motion.
3. State whether you would expect any such model to give a larger, smaller or the same value for the man's speed at C . Give a reason for your answer. A particular model takes account of the resistance by assuming that there is a force of constant magnitude 15 N always acting in the direction opposing the man's motion. This new model also takes account of the man 'pushing off' along the arc from B to C with a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Using an energy method, find the man's speed at C .

1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.

The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).

1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.

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 \(g\).

7 A hollow cylinder, of internal radius 4 m , is fixed so that its axis is horizontal. The point \(O\) is on this axis. A particle, of mass 6 kg , is set in motion so that it moves on the smooth inner surface of the cylinder in a vertical circle about \(O\). Its speed at the point \(A\), which is vertically below \(O\), is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-5_746_739_504_662} When the particle is at the point \(B\), at a height of 2 m above \(A\), find:

1. its speed;
2. the normal reaction between the cylinder and the particle.