A walkthrough for finding g with phyphox and a string: the formula, a T-squared graph, how to cut swing-angle and length errors, and what accuracy to expect.
For g = 9.8 m/s², a pendulum 1.00 m long should swing back and forth in about 2.007 seconds. That figure comes from one line of algebra, and the entire experiment is a matter of working that line backward. Give a phone a period and a string length, and it hands you gravity.
The catch is how much of the answer survives the trip from an app to a number. The sources below agree on the main hazards: swing angle, where the length is measured to, and how many oscillations get timed. This walkthrough takes them in the order they appear in the analysis, finishing with an honest account of how accurate a phone-and-string result can be.
The formula, and the form worth using
For a small swing, the period T of a pendulum of length L is T = 2π√(L/g). The Institute of Physics’ SPARK page gives the same relation and adds the practical rearrangement: plot T² against L and the gradient is 4π²/g.
Solving for g directly gives g = 4π²L/T². One measurement is enough for a number. A graph of several lengths is better, because a straight line through the points exposes a bad reading and averages out the rest. With g = 9.8 m/s², the expected gradient works out to 4π²/9.8, or about 4.03 s² per metre.
Recording the swing
The phyphox wiki’s pendulum experiment reads the phone’s gyroscope, which reports angular velocity that should trace a sine wave. The app finds the period by autocorrelation: it takes the first maximum after the one at zero delay as the period, and its inverse as the oscillation frequency. The IOP page lists the controls as a timed run with the default 3 s start delay and 10 s duration.
- Hang the phone from a string, optionally inside a sealed plastic bag, as the IOP page suggests.
- Measure the length from the pivot to the phone’s centre of mass, not to its end. Balancing the phone on a fingertip locates the centre. Enter the figure in the app.
- Pull the phone to one side, start the recording, release, and keep your hand steady for longer than 13 seconds.
- Keep the swing small. The app’s g calculation assumes it.
Science Buddies offers a second route: tape the phone to the string, record Y-axis acceleration in phyphox, and measure the time between adjacent peaks. That interval is half a period, so double it. If the phone twists as it swings, rest it lightly against a vertical surface such as posterboard, at the cost of a little friction.
Analysing the data
- Note the period phyphox reports for each length. Or time ten full oscillations and divide by ten, a choice Science Buddies makes because timing ten swings beats timing one.
- Use at least five lengths. The eCampusOntario home lab uses string lengths from 30 to 70 cm.
- Square each period and tabulate T².
- Plot T² on the vertical axis against L on the horizontal and draw the best-fit line through the points.
- Take the gradient, then g = 4π² divided by the gradient.
- For uncertainty, follow eCampusOntario: average the period over ten oscillations for each trial and use the standard deviation as its uncertainty.
The table shows what a perfect experiment would produce at g = 9.8 m/s². These numbers are computed from the formula, not measured, and exist so you can see whether your own data are heading the right way.
| Length (m) | Ideal period (s) | Ideal T² (s²) |
|---|---|---|
| 0.30 | 1.099 | 1.209 |
| 0.40 | 1.269 | 1.611 |
| 0.50 | 1.419 | 2.014 |
| 0.60 | 1.555 | 2.417 |
| 0.70 | 1.679 | 2.820 |
A well-behaved graph is a straight line. If the fitted line misses the origin, a likely culprit, reasoning from the formula, is a length measured to the wrong point, since an offset in L shifts every point by the same amount and shows up as an intercept. The gradient is unaffected by a constant offset, which is one more argument for the graph over a single-length calculation.
Points that wander off the line deserve scrutiny before they are discarded. Check the recording for the spiky ends Science Buddies mentions, a swing that grew wide, or a phone that wobbled. Dropping a point is defensible when you can name the cause; dropping it only because it spoils the fit is not.
Cutting the error
Swing angle
The period formula assumes a small swing. HyperPhysics states that the error in the period is less than one percent for amplitudes under 22 degrees. A first-order correction, credited to Bernoulli in an arXiv paper on the pendulum, multiplies the period by 1 + θ²/16, with θ in radians. At 30 degrees that factor is about 1.017, which works out to a g error of roughly 3.4 percent if left uncorrected, since g depends on T². Science Buddies’ 30-degree release angle suits a comparison between lengths, where the same angle repeats every time, but it does not suit an absolute value of g.
Length
Errors in g from the length are direct: a 1 percent error in L is a 1 percent error in g. The phyphox wiki warns that the length must reach the phone’s centre of mass. In a UConn study of large-angle pendulum measurements, a 1 mm length uncertainty alone accounted for 0.09 percent of the total uncertainty.
Wobble and twist
The gyroscope picks up every rotation, so a phone wobbling about other axes adds high-frequency noise. The wiki advises starting the swing without turning the phone. Science Buddies says to ignore the spiky data at the end of a recording, which comes from grabbing the phone.
Timing
A fixed uncertainty in timing shrinks as the number of oscillations grows. With period uncertainty and length uncertainty both known, relative error in g follows from g = 4π²L/T²: it is the relative error in L plus twice the relative error in T. That is a standard propagation rule, not a figure from the sources.
How accurate is a phone pendulum?
No source used here reports a typical result for a phone pendulum, so any promise of a specific percentage would be invented. What can be stated comes from the pieces. A 2 mm slip on a 0.5 m string is 0.4 percent. A 0.01 s timing slip on a 2 s period is 0.5 percent in T and about 1 percent in g. Together, with a small swing and several repeats, a result within a percent or so of expectation is plausible, but that is arithmetic about error sources, not a measured outcome.
At the other end of the scale, careful work is far better. The arXiv paper reports that Nelson and Olsson reached 10⁻⁴ accuracy in g by applying eight or nine correction terms. The same paper also shows an instrument fooling its owner: it attributes a high measured g of 9.963 m/s² to clock-frequency uncertainty, with the clock measured at 6.2 MHz instead of 6.0 MHz.
One more limit sits in the target, not the instrument. g is not the same everywhere. JoVE gives roughly 9.78 m/s² at the equator and 9.83 at the poles, a spread of about 0.05 m/s², or half a percent, caused largely by Earth’s rotation and its flattened shape. A phone pendulum that reads 9.80 at your latitude is not necessarily off by 0.02. Looking up the local value before judging a result keeps you honest.
Related reading
Gravity shows up at planetary scale in x9m8’s explainer on how a Mars flyby gave the Psyche probe a gravity assist, and in the gravitational tug on Earth’s inner core that makes the length of day drift.
Sources
- phyphox wiki, Experiment: Pendulum
- Institute of Physics SPARK, Simple Harmonic Motion: period of a pendulum
- Science Buddies, pendulum project (Phys_p016)
- eCampusOntario, Stay-at-home labs: Simple Pendulum
- HyperPhysics, Large Amplitude Pendulum
- arXiv, Simple Pendulum Revisited
- UConn Physics, Large-angle pendulum measurements (PDF)
- JoVE, Variation in acceleration due to gravity
