2024HSCShort answer

Q25 (a)

2 marks

Question

The mathematical model below shows the relationship between the orbital radius of a satellite and its period.

\frac{r^3}{T^2}=\frac{GM}{4\pi^2}

(a) By considering gravitational force, show how this model can be derived.

Try the question before revealing the answer.

Solution

Criteria	Marks
Shows how the mathematical model is derived	2
Shows some relevant steps	1

Sample answer:

\begin{aligned} F_C &= F_G\\ \frac{mv^2}{r} &= \frac{GMm}{r^2}\\ \left(\frac{2\pi r}{T}\right)^2 &= \frac{GM}{r}\\ \frac{4\pi^2r^2}{T^2} &= \frac{GM}{r}\\ \frac{r^3}{T^2} &= \frac{GM}{4\pi^2} \end{aligned}

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Compare this with your working, not only your final sentence.

Syllabus alignment

Module 5 · Advanced Mechanics

Motion in Gravitational FieldsPredict orbital speed, period, radius and acceleration for planets and satellites, including geostationary orbits.

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