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\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726}
The diagram shows a particle \(P\) of mass 0.5 kg attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity \(6.86 \mathrm {~N} . P\) is released from rest at a point on the surface of the sphere where the acute angle \(A O P\) is at least 0.5 radians.
- (a) For the case angle \(A O P = \alpha , P\) remains at rest. Show that \(\sin \alpha = 2.8 \alpha - 1.4\).
(b) Use the iterative formula
$$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$
with \(\alpha _ { 1 } = 0.8\), to find \(\alpha\) correct to 2 significant figures. - Given instead that angle \(A O P = 0.5\) radians when \(P\) is released, find the speed of \(P\) when angle \(A O P = 0.8\) radians, given that \(P\) is at all times in contact with the surface of the sphere. State whether the speed of \(P\) is increasing or decreasing when angle \(A O P = 0.8\) radians.