 Solution

ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2017-resit/11/c/solution

Let

O

be the original disk centre and

C

the hole centre. The source geometry has

OS = a

OC = CS = a /2

, all collinear. The removed disk has mass

m /4

, since its area is one quarter of the original disk's. Use the parallel axis theorem for both the original disk and the removed disk:

I_{S}^{full} = \frac{1}{2} m a^{2} + m a^{2} = \frac{3}{2} m a^{2}, I_{S}^{hole} = \frac{1}{2} \frac{m}{4} (\frac{a}{2})^{2} + \frac{m}{4} (\frac{a}{2})^{2} = \frac{3 m a ^{2}}{32} .

Subtracting the missing material's moment of inertia gives

I_{S} = \frac{45 m a ^{2}}{32} .

Here

m

continues to denote the original unperforated disk mass; the earring itself has mass

3 m /4

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