1 A particle \(P\) is describing simple harmonic motion of amplitude 5 m . Its speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is 3 m from the centre of the motion. Find, in terms of \(\pi\), the period of the motion.
Find also
- the maximum speed of \(P\),
- the magnitude of the maximum acceleration of \(P\).
\(2 \quad\) A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the lowest point on the inside of a fixed hollow sphere with centre \(O\). The sphere has a smooth internal surface of radius \(a\). Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to \(\frac { 1 } { 2 } u\) the angle \(\theta\) between \(O P\) and the downward vertical satisfies the equation
$$8 g a ( 1 - \cos \theta ) = 3 u ^ { 2 }$$
Find, in terms of \(m , u , a\) and \(g\), an expression for the magnitude of the contact force acting on the particle in this position.