Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 11C / b / ii

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- Solution ii

Solution

Each freely moving particle has position, relative to its own initial radial and tangential unit vectors,

x_{i} (t) = R, e_{r, i} + v t, e_{θ, i} .

Thus all particles lie on a circle of radius

R (t) = R^{2} + v^{2} t^{2} .

Their angular momenta are constant and sum to

L = m R v, \overset{z}{^} .

Since

I (t) = m R (t)^{2}

, the angular velocity is

\dot{θ} = \frac{L}{I ( t )} = \frac{R v}{R ^{2} + v ^{2} t ^{2}} .

With

θ (0) = 0

,

θ (t) = arctan \frac{v t}{R} .

This is the geometry of a freely expanding rotating particle ring.

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