Lighter

This activity allows students to experimentally verify the variation of the acceleration of gravity with altitude. It concretizes the law of universal gravitation and develops the ability to measure small variations.

When you board a commercial aircraft and climb to a cruising altitude of 10,000 meters, you are 10 kilometers farther from the center of the Earth than you were on the ground. According to Newton's law of universal gravitation, the acceleration of gravity decreases with distance from Earth's center as g = GM/r². At sea level, g is about 9.81 m/s²; at 10 km altitude, it decreases to approximately 9.78 m/s². This 0.03 m/s² difference, roughly 0.3%, means you literally weigh less in an airplane. A person weighing 80 kg on the ground would weigh about 76 grams less at cruising altitude, the equivalent of a small apple. While this difference is imperceptible to human senses, the precision accelerometers in modern smartphones can potentially detect it. This experiment challenges students to measure g on the ground and during a flight, compare the values, and verify Newton's gravitational law at two different distances from Earth's center.

Learning objectives:

The student compares the value of the acceleration of gravity measured on the ground with that measured at altitude during an airplane flight. Using the FizziQ accelerometer to record the absolute acceleration in the two situations, the student calculates the difference between the two values then compares it with the theoretical variation predicted by the law of universal gravitation.

Level:

High school

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure the acceleration of gravity at ground level and at aircraft cruising altitude using the smartphone accelerometer
- Compare the two measurements and determine whether the difference is consistent with Newton's law of gravitation
- Calculate the theoretical variation of g with altitude using g = GM/r²
- Understand the relationship between distance from Earth's center and gravitational acceleration
- Appreciate the sensitivity required to detect small variations in g

Scientific concepts:

- Acceleration of gravity
- Law of universal gravitation
- Variation with altitude
- Gravitational field
- Force of gravity

Sensors:

- Accelerometer (absolute acceleration, high-precision static measurement)

What is required:

- Smartphone with the FizziQ application
- A plane flight
- FizziQ experience notebook
- Calculator

Experimental procedure:

Ground measurement: Place the smartphone on a flat, stable surface on the ground (ideally at a known elevation above sea level).
Open FizziQ and select the Accelerometer sensor with absolute acceleration.
Record data for at least 60 seconds while the phone is perfectly still. Calculate the mean and standard deviation.
Repeat the ground measurement three times, recording the mean each time. Calculate the overall average. This is your value of g_ground.
Note your altitude above sea level using a GPS app or known elevation data.
In-flight measurement: During a commercial flight, once the aircraft has reached cruising altitude and the seatbelt sign indicates stable flight, place the smartphone flat on the tray table.
Record data for at least 60 seconds during straight, level flight (avoid turbulence, turns, and altitude changes).
Calculate the mean acceleration. This is your value of g_flight.
Note the cruising altitude from the in-flight display or from a GPS/barometer reading.
Calculate the difference: Δg = g_ground - g_flight.
Calculate the theoretical difference using: Δg = g₀ × (1 - (R/(R+h))²), where R = 6371 km and h is the cruising altitude.
Compare the measured and theoretical differences. Discuss whether the smartphone accelerometer has sufficient precision to detect the altitude effect.
Calculate how much lighter you are (in grams) at cruising altitude compared to on the ground.

Expected results:

The theoretical decrease in g at 10 km altitude is approximately 0.031 m/s² (0.31%). This is at the very limit of smartphone accelerometer precision, which typically has a noise level of 0.02-0.10 m/s² and systematic calibration offsets of 0.05-0.20 m/s². In practice, students may or may not be able to detect the altitude effect, depending on their phone's sensor quality and the stability of the in-flight conditions. Aircraft vibrations, subtle accelerations from flight path adjustments, and the difference in the phone's orientation between ground and flight measurements all introduce errors comparable to the signal being measured. This makes the experiment as much a lesson in measurement limitations as in gravitational physics.

Scientific questions:

- How does Newton's law of gravitation explain the decrease of g with altitude?
- What percentage of your body weight do you lose at cruising altitude?
- Why is it difficult to detect the altitude effect on g with a smartphone accelerometer?
- At what altitude would g be reduced by half compared to its surface value?
- How do precision gravimeters used by geophysicists differ from smartphone accelerometers?
- What other factors (besides altitude) cause local variations in the acceleration of gravity?

Scientific explanations:

The acceleration of gravity g varies with altitude according to Newton's law of universal gravitation: g = G×M/r², where G is the gravitational constant (6.67×10⁻¹¹ m³kg⁻¹s⁻²), M the mass of the Earth (5.97×10²⁴ kg), and r the distance from the center of the Earth. At the Earth's surface (r ≈ 6371 km), g is approximately 9.81 m/s².

At the cruising altitude of a commercial aircraft (10-12 km), the theoretical reduction in g is approximately 0.003 m/s² (or 0.03% of its ground value). The smartphone's accelerometer measures "absolute acceleration", which corresponds to the standard of the acceleration vector including gravity.

At rest on a horizontal surface, it reads approximately 9.81 m/s². This measurement is possible because the sensor detects the reaction force of the support (normal force) which exactly opposes gravity.

In horizontal flight at constant speed, the plane is in balance: the lift exactly compensates for the weight. The accelerometer therefore continues to measure "apparent gravity".

The decrease in g with altitude is a subtle phenomenon, at the limit of the sensitivity of smartphone accelerometers (±0.01 m/s²). To optimize the measurement, it is recommended to: 1) Take long recordings (>10 seconds) and calculate the average to reduce noise; 2) Ensure that the aircraft is in stable flight (no turbulence or maneuvers); 3) Place the smartphone on a horizontal surface.

Even with these precautions, the measurement remains delicate and can be affected by other factors such as aircraft vibrations. This experiment nevertheless illustrates a fundamental principle: gravity is not constant but decreases with altitude, a reality taken into account in calculations of space mechanics and precision geodesy.

Extension activities:

Frequently asked questions:

Q: I cannot tell whether the in-flight measurement is different from the ground measurement. Does this mean the experiment failed?
R: Not at all. The inability to detect the difference is itself an important scientific result: it tells you that the smartphone's measurement precision is not sufficient for this particular measurement. Understanding the limits of your instruments is a core experimental skill.

Q: The in-flight readings are much noisier than the ground readings. Why?
R: Aircraft vibrations from engines and air turbulence produce continuous small accelerations that add noise to the measurement. Even in smooth flight, the aircraft makes constant small adjustments. Take long recordings and use averaging to reduce this noise.

Q: Can I do this experiment on a mountain instead of in an airplane?
R: At 2000 m altitude, the theoretical decrease in g is only about 0.006 m/s², which is even harder to detect than the airplane case. However, a mountain offers a vibration-free environment, which may partially compensate for the smaller signal.

Q: Why do I need to use the same smartphone for both measurements?
R: Different smartphones have different calibration offsets (systematic errors) in their accelerometers. The difference between two phones can easily be 0.1-0.3 m/s², much larger than the gravitational signal. Using the same phone ensures that calibration errors cancel when computing the difference.