Elastic string ring or sphere surface

A particle attached to an elastic string moves on the surface of a fixed sphere or through a ring, requiring energy conservation combining elastic PE, gravitational PE, and kinetic energy to find speed or verify contact conditions.

1 questions · Challenging +1.3

6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.05e Radial/tangential acceleration
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OCR M3 2012 January Q7
15 marks Challenging +1.3
7 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-4_351_314_255_861} One end of a light elastic string, of natural length \(\frac { 2 } { 3 } R \mathrm {~m}\) and with modulus of elasticity 1.2 mgN , is attached to the highest point \(A\) of a smooth fixed sphere with centre \(O\) and radius \(R \mathrm {~m}\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string and is in contact with the surface of the sphere, where the angle \(A O P\) is equal to \(\theta\) radians (see diagram).
  1. Given that \(P\) is in equilibrium at the point where \(\theta = \alpha\), show that \(1.8 \alpha - \sin \alpha - 1.2 = 0\). Hence show that \(\alpha = 1.18\) correct to 3 significant figures. \(P\) is now released from rest at the point of the surface of the sphere where \(\theta = \frac { 2 } { 3 }\), and starts to move downwards on the surface. For an instant when \(\theta = \alpha\),
  2. state the direction of the acceleration of \(P\),
  3. find the magnitude of the acceleration of \(P\).