Read the guide: Vibration frequency explained
How to run the pendulum experiment
-
Step 1: Tie about 1 m (3 ft) of strong, thin string to a sturdy zip-top bag or small cloth bag with a double knot. Tie the other end to something fixed, like a hook or a broom handle laid across two chairs.
-
Step 2: Hang it low over a bed or cushion, clear of people, walls and furniture.
-
Step 3: Measure the length from the pivot point to where the middle of the phone will sit in the bag. Enter it in the tool. Record for is 20 seconds by default, which suits a 1 m string. For a longer string, set it to 30 seconds or more.
-
Step 4: Tap Start. On iPhone or iPad, tap Allow when Safari asks for Motion & Orientation access. Put the phone in the bag, seal it, and let it hang still. The tool waits 10 seconds (Start delay) before it records, which gives you time for this step and the next. If you need longer, pick 20 s before you tap Start.
-
Step 5: Pull the bag about 10 cm (4 in) to one side, keeping the string tight, and let go. Don't push it.
-
Step 6: Let it swing for at least 10 full swings, about 20 to 30 seconds on a 1 m string. Then read the period, the frequency and your value of g.
This pendulum frequency experiment turns your phone into the weight on the end of a string. The phone’s rotation sensor feels every swing, so the page times the period for you, with no stopwatch and no reaction-time error. From the period and the string length, it works out g, the acceleration due to gravity.
The physics behind the pendulum frequency experiment
A pendulum swings at a steady rate set by its length. The time for one full swing, there and back, is the period T, in seconds. The frequency f is how many full swings happen each second: f = 1 ÷ T, in hertz (Hz).
For small swings, the period is T = 2π√(L ÷ g). Rearrange it to get g from what you measured:
g = 4π²L ÷ T²
L is the length in meters, from the pivot to the middle of the phone. T is the period in seconds.
Worked example
You measure L = 0.80 m and the tool reports T = 1.80 s.
g = 4 × 3.1416² × 0.80 ÷ 1.80² = 31.58 ÷ 3.24 = 9.75 m/s²
That’s within 1% of the standard 9.81 m/s². The frequency is 1 ÷ 1.80 = 0.56 Hz, about 33 swings per minute.
Periods to expect
Worked out with g = 9.81 m/s²:
| Length L | Period T | Frequency |
|---|---|---|
| 0.25 m | 1.00 s | 1.00 Hz |
| 0.50 m | 1.42 s | 0.70 Hz |
| 0.75 m | 1.74 s | 0.58 Hz |
| 1.00 m | 2.01 s | 0.50 Hz |
| 1.50 m | 2.46 s | 0.41 Hz |
| 2.00 m | 2.84 s | 0.35 Hz |
Your phone takes about 60 readings a second, far faster than any of these swings, so it has plenty of data for each one.
Why small swings matter
The formula assumes small swings. Wider swings take a little longer, so g comes out too low:
- At 10° the period is about 0.2% longer, so g reads about 0.4% low.
- At 20° it’s about 0.8% longer, so g reads about 1.5% low.
- At 30° it’s about 1.7% longer, so g reads about 3.4% low.
Keep the swing under about 10°. On a 1 m string, that means pulling the bag no more than about 15 cm (6 in) to the side.
Common causes and fixes
If your value of g is off by more than a few percent, check these.
- Wrong length. Measure to the middle of the phone, not to the knot or the bottom of the bag. Measure again before you blame the phone.
- Swing too wide. Start with a smaller pull, as shown above.
- Bag twisting or circling. A spinning bag or an oval path gives the sensor a messy signal. Let it hang still, pull it straight back, and let go without a push.
- Stretchy string or a wobbly support. Elastic cord, yarn and a loose hook all change the length during the swing. Use thin, strong string tied to something solid.
- String too short. On a short string, the size of the phone matters and the simple formula fits less well. Use at least 0.5 m. A meter or more is better.
- Too few swings. Let it run for at least 10 full swings so small errors average out.
Ideas for teachers and students
- Change the length, for example 0.4, 0.6, 0.8 and 1.0 m. Plot T² against L. You should get a straight line through zero with a slope of 4π² ÷ g.
- Keep the length the same and add a small weight to the bag. The period barely changes, which shows that mass doesn’t matter.
- Compare a 5° swing with a 30° swing to see the small-angle effect for yourself.
- Compare your answer with 9.81 m/s². Real g varies from about 9.78 m/s² at the equator to about 9.83 m/s² at the poles, but that’s smaller than most classroom errors.
Use the worksheet download under the tool to record each group’s lengths, periods and results.
What this tool can’t measure
- Wide or tangled swings. Past about 30°, or when the bag spins, results get less reliable.
- Lab-grade g. Treat the answer as a good guide, not a certified measurement. A careful length measurement matters more than the phone.
- Phones without a rotation sensor. The tool times the swing with the accelerometer instead, which is a little less reliable. Check yours with the accelerometer test.
To learn how period, frequency and RPM relate, read vibration frequency explained.
Frequently asked questions
Does the weight of the phone change the result?
No. The period of a simple pendulum depends only on its length and on g, not on the mass. A heavy phone swings at the same rate as a light one on the same string, as long as you measure the length to its middle.
How close to 9.81 m/s² should I get?
With a careful length measurement and small swings, you can usually get within a few percent. Most of the error comes from the length, wide swings and a twisting bag. A 1 cm length error on a 1 m string changes g by about 1%.
Why does the tool use the rotation sensor?
As the phone swings, it tilts back and forth with the string, so its rotation sensor sees one clean wave per swing. The accelerometer reading along the string peaks each time the phone passes the bottom, which is twice per swing. The rotation signal is easier to time without mistakes.
Is it safe to swing my phone?
It is if you set it up with care. Use strong string and a sealed bag, tie double knots, hang it low over something soft, and keep swings small. Never spin it around or swing it near people or glass.
Can I use this in class?
Yes. Each group needs only a phone, a bag, string and a tape measure. Groups can use different lengths and compare results. Nothing is uploaded and there's no signup.
Last updated .