Latitude and gravity

This activity allows students to verify that the acceleration of gravity varies depending on the Earth's latitude. It recreates an important historical discovery with modern tools.

In 1672, the French astronomer Jean Richer sailed from Paris to Cayenne in French Guiana carrying a precision pendulum clock. He was astounded to discover that the clock, perfectly adjusted in Paris, lost 2.5 minutes per day near the equator. This seemingly small discrepancy had enormous implications: it meant that the acceleration of gravity was weaker at the equator than at higher latitudes. Newton later explained this variation as the combined effect of two factors: the Earth is not a perfect sphere but an oblate ellipsoid (bulging at the equator), so the surface is farther from the center at the equator; and the Earth's rotation produces a centrifugal effect that partially counteracts gravity, more strongly at the equator where the rotational speed is highest. The total variation amounts to about 0.5% of g, from roughly 9.78 m/s² at the equator to 9.83 m/s² at the poles. Modern smartphone accelerometers are just barely precise enough to detect this difference, making it possible to recreate Richer's discovery by comparing measurements from different latitudes.

Activity overview:

The student measures the acceleration of gravity using the FizziQ accelerometer in a static position on a table. By calculating the average of the recorded values and comparing this result with that of a class located near the equator, the student can verify experimentally that g is lower at the equator than at mid-latitudes in accordance with the historical observation of the astronomer Richer.

Level:

High school

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure the acceleration of gravity at your location using the smartphone accelerometer
- Compare your measurement with values from other latitudes (partner schools or published data)
- Understand the two physical causes of g variation with latitude: Earth's shape and rotation
- Calculate the theoretical value of g at your latitude and compare with the measurement
- Appreciate the historical significance of latitude-dependent gravity in shaping our understanding of Earth's shape

Scientific concepts:

- Acceleration of gravity
- Shape of the Earth
- Centrifugal force
- Scientific collaboration
- History of science

Sensors:

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

Material needed:

- Smartphone with the FizziQ application
- A flat and stable surface
- Contact with a class located near the equator (ideally)
- Internet access to check official g values
- FizziQ experience notebook

Experimental procedure:

Open FizziQ and select the Accelerometer sensor. Choose absolute acceleration for the most precise measurement.
Place the smartphone on a perfectly flat, stable surface (a solid table, not a desk that flexes). Ensure the surface is level.
Record the acceleration for at least 60 seconds at the maximum sampling rate. Minimize all environmental vibrations: close doors, avoid walking near the phone.
Calculate the average (mean) value and the standard deviation of the recorded data. Record these values.
Repeat the measurement three times, removing and replacing the phone each time. Calculate the overall mean and standard deviation.
Determine your geographic latitude using a GPS app or online map service. Record it in degrees.
Look up the theoretical value of g at your latitude using the International Gravity Formula: g = 9.7803267715 × (1 + 0.0053024 × sin²φ - 0.0000058 × sin²2φ), where φ is the latitude.
Compare your measured mean with the theoretical prediction. The difference represents the combined systematic error of the sensor plus local gravity anomalies.
If possible, obtain a measurement from a partner school at a different latitude (ideally near the equator or at a significantly different latitude). Compare the two measurements.
Calculate the expected difference between the two latitudes using the gravity formula. Is it consistent with the measured difference?
Discuss the minimum detectable latitude difference given the precision of your smartphone accelerometer.
Research Jean Richer's historical measurement and compare his observed discrepancy with your theoretical calculation for Paris versus Cayenne.

Expected results:

At a mid-latitude location (e.g., Paris at 48.9°N), the theoretical value of g is approximately 9.809 m/s². At the equator (0°), it is approximately 9.780 m/s². The difference of about 0.029 m/s² (0.3%) is small but potentially detectable with a good smartphone accelerometer. However, individual smartphone accelerometers typically have systematic calibration offsets of 0.05-0.20 m/s², which may exceed the latitude-dependent variation. Comparing two different smartphones at the same location will show differences due to calibration, not latitude. The most convincing approach is to use the same phone at two different latitudes, though this requires travel. Students should appreciate that detecting a 0.03 m/s² difference requires measurement precision better than 0.01 m/s², which is at the limit of consumer accelerometers.

Scientific questions:

- Why is the acceleration of gravity smaller at the equator than at the poles?
- What are the two main physical causes of this variation, and which one contributes more?
- How did Richer's observation help confirm that the Earth is not a perfect sphere?
- What is the centrifugal effect, and why is it strongest at the equator?
- How precise would a sensor need to be to detect the difference in g between the ground floor and the top of a tall building?
- How do geophysicists measure local variations in g, and what do these variations reveal about underground geology?

Scientific explanations:

In 1672, the astronomer Jean Richer made a surprising discovery during an expedition to Cayenne (French Guiana): a pendulum clock, perfectly adjusted in Paris, was delayed by 2.5 minutes per day near the equator. This historical observation revealed that the acceleration of gravity g varies with latitude.

This variation is explained by two main factors: 1) The ellipsoidal shape of the Earth (flattened at the poles): the Earth's radius is shorter at the poles (6357 km) than at the equator (6378 km), which brings the poles closer to the Earth's center of mass, thus increasing the gravitational force; 2) The centrifugal force due to the earth's rotation: maximum at the equator and zero at the poles, it partially opposes gravity. These two effects combined cause g to vary from approximately 9.78 m/s² at the equator to 9.83 m/s² at the poles, a difference of 0.5%.

The accelerometer of a modern smartphone is sensitive enough to detect this variation, although individual instrumental errors may be of the same order of magnitude. This is why this experiment benefits from a collaborative approach: by averaging the measurements of several devices, random errors partially cancel each other out.

Historically, this discovery helped confirm Newton's theory of the shape of the Earth and led to the first geodetic expeditions to precisely measure the shape of the Earth, notably that of Maupertuis in Lapland (1736) and that of La Condamine in Peru (1735-1744).

Extension activities:

Frequently asked questions:

Q: My measured value of g does not match the theoretical value for my latitude. Is my phone broken?
R: Smartphone accelerometers have systematic calibration offsets that typically range from 0.05 to 0.20 m/s². This is larger than the latitude-dependent variation. The experiment is more meaningful when comparing relative differences (same phone at different locations) rather than absolute values.

Q: Can I detect the latitude effect by comparing measurements from two different smartphones?
R: Unfortunately, no. The calibration difference between two smartphones (typically 0.1-0.2 m/s²) is much larger than the latitude effect (0.03 m/s²). You need to use the same phone at two significantly different latitudes.

Q: How much latitude difference do I need to detect the effect?
R: With a well-calibrated smartphone (precision ±0.02 m/s²), you would need a latitude difference of at least 15-20 degrees to reliably detect the g variation. Comparing Paris (49°N) with a tropical location (0-10°N) should show a measurable difference of about 0.02-0.03 m/s².

Q: Why does the Earth's rotation reduce the measured g at the equator?
R: On a rotating Earth, objects at the surface follow a circular path and require centripetal acceleration directed toward the Earth's axis. This centripetal acceleration is provided by a small fraction of the gravitational pull, leaving less to be measured as apparent gravity. The effect is maximum at the equator where the rotational speed is highest (about 465 m/s).