3 Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10 ^ { 8 } \mathrm {~km}\) at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10 ^ { 24 } \mathrm {~kg}\),

1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons,
2. state the direction in which this force acts.
   \(4 \quad A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(A P B = 90 ^ { \circ }\) (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-3_286_745_402_660} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda _ { A } \mathrm {~N}\). The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda _ { B } \mathrm {~N}\).
3. (a) Show that \(\lambda _ { B } = \frac { 8 } { 3 } \lambda _ { A }\).
   (b) Find an expression for \(\lambda _ { A }\) in terms of \(m\) and \(g\).
4. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3 m \mathrm {~kg}\) is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.
5. Find the distance of \(Q\) below \(B\) in this equilibrium position.