CIRCULAR MOTION

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The axis of a smooth fixed cylinder of radius R is horizontal. Two particles P1, of mass m, and P2, of mass 2m, are joined together by a model string. P1 is initially at one end B of the horizontal diameter AB, and P2 hangs vertically below the other end A. O is the centre of the vertical cross-section. At time t = 0 the system is released from rest, and at time t the particle P1 has rotated through an angle θ, as shown below.
(a) For each particle, define an appropriate coordinate system and draw a force diagram showing all the forces acting on each particle at time. Define any variables or parameters that you introduce.

(b) Explain why it is better to use plane polar vectors for particle P1.Include them on your force diagram, and express the acceleration of the particle in terms of these unit vectors.

(c) Derive the equation of motion for particle P1, and express this equation in component form.

(d) Write down the equation of motion for particle P2.

(e) Write down the distance through which P2 has fallen after time t, and hence deduce that the downward component of the acceleration of P2 at this time is R¨θ, where ¨θ is the component of the angular acceleration of particle P1.

(f) Use the results of parts (c)–(e) and the property of a model string to show that ¨θ = (g/3R) * (2 − cos θ)

(g) Determine ˙θ^2 as a function of θ, and hence show that the magnitude of the normal reaction of the cylinder on P1 is (mg/3) * (5 sin θ − 4θ)