6
\includegraphics[max width=\textwidth, alt={}, center]{e004bbb5-f9e1-4ea2-8357-39db9392cb8c-4_559_525_258_808} A particle \(P\) of weight 6 N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 0.8 m . The string has natural length \(\frac { 1 } { 10 } \pi \mathrm {~m}\) and modulus of elasticity \(9 \mathrm {~N} . P\) is released from rest at a point \(X\) on the sphere where \(O X\) makes an angle of \(\frac { 1 } { 4 } \pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(O P\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac { 1 } { 6 } \pi , P\) passes through the point \(Y\).

1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2 \cdot 4 ( \sqrt { 3 } - \sqrt { 2 } ) \mathrm { J }\) and the elastic potential energy in the string decreases by \(0.4 \pi \mathrm {~J}\).
2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac { 1 } { 6 } \pi\), and hence find the maximum speed of \(P\).