4 The diagram shows two particles P and Q , of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m . Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to \(\theta ^ { \circ }\). The pulley, P and Q all lie in the same vertical plane.
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Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P .
- Given that P is in equilibrium at the point where \(\theta = \alpha\), determine the value of \(\alpha\).
Particle P is now released from rest at the point on the surface where \(\theta = 35\), and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.
- By considering energy, determine the speed of P at the instant when \(\theta = 45\).
- State one modelling assumption you have made in determining the answer to part (b).