String through hole – hanging particle in equilibrium below table

3. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string of length 6l. The string passes through a small smooth fixed ring at the point \(A\). The particle \(Q\) is hanging freely at a distance \(l\) vertically below \(A\). The particle \(P\) is moving in a horizontal circle with constant angular speed \(\omega\). The centre \(O\) of the circle is vertically below \(A\). The particle \(Q\) does not move and \(A P\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. Show that

1. \(\theta = 60 ^ { \circ }\)
2. \(\omega = \sqrt { } \left( \frac { 2 g } { 5 l } \right)\)

8 A particle, \(P\), of mass 3 kg is attached to one end of a light inextensible string. The string passes through a smooth fixed ring, \(O\), and a second particle, \(Q\), of mass 5 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 \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, as shown in the diagram. The angle between \(O P\) and the vertical is \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-5_474_476_1425_774}

1. Explain why the tension in the string is 49 N .
2. Find \(\theta\).
3. Find the radius of the horizontal circle.

5 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes through a hole in a smooth horizontal table. The edges of the hole are also smooth. Particle \(A\), of mass 1.4 kg , moves, on the table, with constant speed in a circle of radius 0.3 m around the hole. Particle \(B\), of mass 2.1 kg , hangs in equilibrium under the table, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-4_684_1022_1176_504}

1. Find the angular speed of particle \(A\).
2. Find the speed of particle \(A\).
3. Find the time taken for particle \(A\) to complete one full circle around the hole.

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 \(\omega\).

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]