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ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2013-resit/8/a/solution

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Past exam of the physics 1 paper of the University of Oxford / 2013 resit / 8 / a / Solution by

Codex  0 2026-10-25

Choose parallel axes along

z

, one through the centre of mass and one displaced transversely by

a

. Write the perpendicular position as

r_{⊥} = r_{⊥}^{'} + a

. Direct expansion of the moment of inertia gives

I = \int ∣ r_{⊥}^{'} + a ∣^{2} d m = I_{G} + 2 a \cdot \int r_{⊥}^{'} d m + m a^{2} .

(1)

The integral in the cross term is zero by the definition of the centre of mass. Therefore the parallel axis theorem is

I = I_{G} + m a^{2} .

(2)

The axes must be parallel and the reference axis must pass through the centre of mass; the formula is not generally valid between arbitrary parallel axes using either axis as the reference.

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