3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) down the plane, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 \mathrm { e } ^ { - x } \mathrm {~N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 2 \mathrm { e } ^ { - x }\).
2. Find \(v\) when \(x = 0.6\).
   \includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-3_905_604_251_769} A uniform solid cone has base radius 0.4 m and height 4.4 m . A uniform solid cylinder has radius 0.4 m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal (see diagram).
3. Calculate the least possible value of the coefficient of friction between the plane and the object.
4. Calculate the greatest possible height of the cylinder.