Coupled circular motions

7 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).

1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
   1. the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
   2. the speeds of the particles.

The radius of the Earth is \(R\) m. The force of attraction towards the centre of the Earth experienced by a body of mass \(m\) kg at a distance \(x\) m from the centre is of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant.

1. Show that \(k = gR^2\). [1 mark]

Two satellites \(A\) and \(B\), each of mass \(m\) kg, are moving in circular orbits around the Earth at distances \(3R\) m and \(4R\) m respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,

1. show that the ratio of the speed of \(B\) to that of \(A\) is \(4:3\). [2 marks]

If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,

1. find, in terms of \(m\) and \(g\), the magnitude of the force in the wire. [8 marks]

\includegraphics{figure_6} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg, by three light rods where the length of rod \(AP\) is 1.5 m and the length of rod \(PQ\) is 0.75 m. Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A\), \(B\), \(P\) and \(Q\) are coplanar. The rod \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).

1. Find the tension in the rod \(PQ\). [2]
2. Find \(\omega\). [3]
3. Find the speed of \(P\). [1]
4. Find the tension in the rod \(AP\). [3]
5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]