 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ia/paper-4/12c/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Write

r_{i} = X + r_{i}^{'}

and

v_{i} = V + v_{i}^{'}

, where

V = \dot{X}

\sum_{i} m_{i} r_{i}^{'} = 0, \sum_{i} m_{i} v_{i}^{'} = 0.

Expand the total angular momentum:

L = i \sum m_{i} (X + r_{i}^{'}) \times (V + v_{i}^{'}) = M X \times V + X \times i \sum m_{i} v_{i}^{'} + (i \sum m_{i} r_{i}^{'}) \times V + i \sum m_{i} r_{i}^{'} \times v_{i}^{'} .

L = M X \times V + i \sum m_{i} r_{i}^{'} \times v_{i}^{'} .

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