 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v-resit/question-7/b/ii/solution

x^{'} = s cos θ

y^{'} = s sin θ

0 \leq s \leq a

and

0 \leq θ \leq π

s d s d θ

, and the separation from

(0, 0, h)

s^{2} + h^{2}

. Thus

ϕ (h) = - G σ \int_{0}^{π} \int_{0}^{a} \frac{s d s d θ}{s ^{2} + h ^{2}} = - π G σ [s^{2} + h^{2}]_{0}^{a} = - π G σ (a^{2} + h^{2} - h), h > 0.

This Newtonian gravitational potential is half that of the full disk because the separation is independent of the polar angle. At large

h

it approaches

- G (π a^{2} σ /2) / h

, with the correct total mass.

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