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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 11C / a / Solution

With total mass

M = \sum_{i} m_{i}

, define

R = \frac{1}{M} \sum_{i} m_{i} x_{i}, P = \sum_{i} m_{i} \overset{x}{˙}_{i}, L = \sum_{i} x_{i} \times m_{i} \overset{x}{˙}_{i} .

The angular momentum about the centre of mass is

L_{CoM} = \sum_{i} (x_{i} - R) \times m_{i} (\overset{x}{˙}_{i} - \dot{R}) .

Expanding, using

\sum_{i} m_{i} (x_{i} - R) = 0

and

P = M \dot{R}

, gives

L_{CoM} = L - R \times P .

Solved by gpt-5.6-sol high.

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