Maximum/minimum tension or reaction

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can rotate freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\). The line \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), where \(\alpha < \frac { \pi } { 2 }\) When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 4.

1. Show that \(v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )\) Given that \(\cos \alpha = \frac { 2 } { 5 }\) and that \(u = \sqrt { 3 g l }\)
2. show that \(P\) moves in a complete vertical circle. As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(k T\)
3. Find the exact value of \(k\)

4. \begin{figure}[h] \end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.

1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac { \pi } { 2 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.

1. Show that \(v ^ { 2 } = u ^ { 2 } + 2 g l ( \cos \alpha - \cos \theta )\). It is given that \(\cos \alpha = \frac { 3 } { 5 }\) and that \(P\) moves in a complete vertical circle.
2. Show that \(u > 2 \sqrt { } \left( \frac { g l } { 5 } \right)\). As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5 T\).
3. Show that \(u ^ { 2 } = \frac { 33 } { 10 } \mathrm { gl }\).

7. \begin{figure}[h] \end{figure} 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 \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P , \angle A O P = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(O A\) where \(\cos \alpha = \frac { 2 } { 3 }\).

1. Find \(u\) in terms of g and \(l\). When \(\angle A O P = \theta\), the tension in the string is \(T\).
2. Show that \(T = \frac { m g } { 3 } ( 9 \cos \theta - 4 )\).
3. Find the range of values of \(T\). END

6 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A small bead, of mass \(m\), is attached to the other end of the string. The bead is moving in a vertical circle, centre \(O\). When the bead is at \(B\), vertically below \(O\), the string is taut and the bead is moving with speed \(5 v\).
\includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_536_554_502_774}

1. The speed of the bead at the highest point of its path is \(3 v\). Find \(v\) in terms of \(a\) and \(g\).
2. Find the ratio of the greatest tension to the least tension in the string, as the bead travels around its circular path.
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_1261_1709_1446_153}

8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).

1. Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
   (4 marks)
2. \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760} Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).

5 An item of clothing is placed inside a washing machine. The drum of the washing machine has radius 30 cm and rotates, about a fixed horizontal axis, at a constant angular speed of 900 revolutions per minute. Model the item of clothing as a particle of mass 0.8 kg and assume that the clothing travels in a vertical circle with constant angular speed. Find the minimum magnitude of the normal reaction force exerted by the drum on the clothing and find the maximum magnitude of the normal reaction force exerted by the drum on the clothing.
[0pt] [6 marks]

2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length \(a\) metres, and is moving in a vertical circle with centre O and radius \(a\) metres. When P is vertically below O , its speed is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When OP makes an angle \(\theta\) with the downward vertical, and the string is still taut, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the tension in the string is \(T \mathrm {~N}\), as shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find an expression for \(v ^ { 2 }\) in terms of \(a\) and \(\theta\), and show that $$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
2. Given that \(a = 0.9\), show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
3. Find the largest value of \(a\) for which P moves in a complete circle.
4. Given that \(a = 1.6\), find the speed of P at the instant when the string first becomes slack.