 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/14/a/i/solution

d F = \frac{G m _{2}}{r ^{2}} d m_{1} + \frac{G m _{1}}{r ^{2}} d m_{2} - \frac{2 G m _{1} m _{2}}{r ^{3}} d r .

Dividing by the positive value

F

gives the particularly useful relative form

\frac{d F}{F} = \frac{d m _{1}}{m _{1}} + \frac{d m _{2}}{m _{2}} - 2 \frac{d r}{r} .

For finite small changes this gives the first-order approximation to the fractional change; higher-order products of increments are omitted.

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