2024HSCShort answer

Q25 (b)

4 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}

(b) A planet with five moons is discovered. The following graph is produced from observations of the orbital radius of the moons and their orbital periods, measured in Earth days.

Use the graph to calculate the mass of the planet.

Uncropped source page 25 for question 25.b; requires human review and cropping — Reviewed visual · source page 25

Try the question before revealing the answer.

Solution

Criteria	Marks
Calculates the mass of the planet using the graph’s line of best fit and correct working	4
Shows some steps in calculating the gradient; relates the gradient to the mathematical model and/or mass of the planet	3
Shows some steps in calculating the gradient OR relates the graph to the mathematical model and/or mass of the planet	2
Provides some relevant information	1

Sample answer: Using the line of best fit

\text{Gradient} = 0.0008 \times \frac{\mathrm{days}^2}{10^{12}\ \mathrm{km}^3} = \frac{0.0008 \times (24 \times 60 \times 60)^2}{10^{12} \times 1000^3}\ \mathrm{s^2m^{-3}} = 6.0 \times 10^{-15}\ \mathrm{s^2m^{-3}}

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

6.0 \times 10^{-15} = \frac{4\pi^2}{6.67 \times 10^{-11} \times M}

M = 9.9 \times 10^{25}\ \mathrm{kg}

Supplied marking guidance · human reviewed

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.

Spot a problem? Send feedback