Quick answer: Gravitational potential energy is U = −GMm/r, with zero at infinity. This AP Physics C lab on orbits and escape speed gives the aim, the equipment, the safety points, a step-by-step method, sample results and 7 analysis questions; the answers are on a separate page and the lab sheet is a free printable PDF.
Do the moons of a giant planet obey the same rule as the planets around the Sun? This lab uses a table of moon radii and periods to test T² ∝ r³ and to turn the gradient of the graph into a measurement of the mass of a planet that nobody can put on a balance.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:M Unit 7
Aim
To test Kepler’s third law with orbital data for four moons of one planet, and to use the result to find the planet’s mass.
Equipment
- The data table below (moon radius and period)
- Graph paper or a spreadsheet
- A calculator with a power key
- A ruler for the line of best fit
Safety
This is a desk-based data lab. Sit with the screen and paper at a comfortable height, take a break from the screen every 20 minutes, and keep drinks away from any computer.
Method
- Convert each period from days to seconds (1 day = 86 400 s).
- Calculate T² and r³ for each moon and complete the table.
- Plot T² on the vertical axis against r³ on the horizontal axis, with the origin included.
- Draw the line of best fit and find its gradient using the first and last points.
- Use the gradient to calculate the mass of the planet from gradient = 4π²/GM.
Results
Sample data, with the calculated columns already filled for checking:
| Moon | r (10⁸ m) | T (days) | T (10⁵ s) | T² (10¹⁰ s²) | r³ (10²⁶ m³) |
|---|---|---|---|---|---|
| Io | 4.22 | 1.77 | 1.529 | 2.34 | 0.752 |
| Europa | 6.71 | 3.55 | 3.067 | 9.41 | 3.02 |
| Ganymede | 10.70 | 7.15 | 6.178 | 38.2 | 12.3 |
| Callisto | 18.83 | 16.69 | 14.42 | 208 | 66.8 |
Analysis questions
Question 1 Unit conversion
Convert the period of Ganymede to seconds and calculate T² and r³ for it.Question 2 Shape of the graph
Describe the graph of T² against r³ and state what it shows about Kepler’s third law.Question 3 Gradient
Calculate the gradient using the Io and Callisto points, with its unit.Question 4 Mass of the planet
Use the gradient to calculate the mass of the planet.Question 5 Logarithmic test
Show using Io and Callisto that log T plotted against log r has gradient 1.5.Question 6 Uncertainty
Each period is given to three significant figures. Estimate the percentage uncertainty in the planet’s mass.Question 7 Assumptions
State two assumptions of the method and how they could affect the result.
