Q: What does Measuring g Using a Smartphone Pendulum for A-Level Physics Practicals cover? A: Transform your phone into a precision timer for pendulum experiments.
TL;DR Your smartphone's accelerometer can time pendulum swings with millisecond-level timestamps when the app samples fast
enough. This guide shows how to measure g within a few percent of 9.81m/s2, handle
amplitude/systematic effects, and extract clean data using free sensor apps-good rehearsal for H2 Physics practical prep
or home revision.
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Traditional pendulum experiments suffer from human reaction time (~0.2s), which can push
g
a few percent off. Many
phones sample accelerometer data at 100 - 400 Hz (check your app/device), detecting the exact moment of direction change at
each swing endpoint.
Key advantages:
Precision timing: millisecond timestamping vs 0.01 s on most stopwatches
Automated counting: No losing track after 20 oscillations
Raw data export: Direct to spreadsheet for instant analysis
Multiple sensors: Cross-validate with gyroscope data
Setting Up Your Smartphone Pendulum
Materials Needed
Smartphone with accelerometer (any phone from 2015+)
String or fishing line (1-2m)
Small pouch/sock to hold phone securely
Tape measure or metre rule
Protractor or angle-measuring app
Retort stand or ceiling hook
Step-by-Step Setup
Download a sensor app:
Android: "Physics Toolbox Sensor Suite" or "Phyphox"
iOS: "Sensor Logger" or "SensorLog"
Secure the phone:
Place in small pouch/sock
Ensure phone can't slip during swings
Keep centre of mass aligned with string attachment
Measure pendulum length L:
From pivot point to phone's centre of mass
Typical good range: 0.8-1.5m
Record uncertainty: ±0.002m
Set initial amplitude:
Start with small angles (5-10°)
Use protractor at release point
Mark position with tape for consistency
Data Collection Protocol
Recording Clean Data
Start sensor recording before releasing pendulum
Release gently (no initial velocity)
Record 20-30 complete oscillations
Keep amplitude small (under 10°)
Stop recording after pendulum settles
What Your Data Shows
The accelerometer measures total acceleration including gravity. During a swing:
At endpoints: Max acceleration (direction change)
At equilibrium: Min acceleration (max velocity)
Period T: Time between acceleration peaks
Extracting the Period
Method 1: Peak Counting
Export data to spreadsheet
Plot acceleration vs time
Count peaks over 20 oscillations
Calculate: T=20total time
Method 2: Fourier Analysis (Advanced)
Use FFT function in spreadsheet
Identify fundamental frequency f
Period T=f1
Calculating g
For small angles, the simple pendulum formula applies:
T=2πgL
Rearranging for g:
g=T24π2L
Error Analysis That Scores Full Marks
Systematic Errors
Finite amplitude correction:
Formula assumes θ→0
For small angle θ0 in radians: Tactual=Ttheory(1+16θ02)
Keep θ0<10° for less than one-tenth of a percent error
Air resistance:
Causes amplitude decay
Use first 10 swings for cleanest data
Or fit exponential decay and extrapolate
Finite mass distribution:
Phone isn't a point mass
Measure to centre of mass carefully
Error typically stays below one percent if L>1m
Random Errors
Length measurement: δL=±0.002m
Period determination: set δT to half your sampling interval e.g.,200Hz≈±0.0025s
Propagate to g: Result:gδg=(LδL)2+(T2δT)2
Graphical Analysis Methods
The T2 vs L Plot
Instead of calculating g from single measurements:
Vary pendulum length (0.5m to 1.5m in 0.1m steps)
Measure period for each length
Plot T2 against L
Gradient = g4π2
Calculate: g=gradient4π2
This method:
Reduces random errors through multiple points
Reveals systematic issues (non-zero intercept)
Shows measurement quality (R² value)
Common Exam Variations
1. "Effect of Amplitude on Period"
Measure T for angles 5°, 10°, 15°, 20°
Plot T vs θ02
Verify quadratic relationship
Extract g from intercept
2. "Compound Pendulum"
Phone's extended mass matters more
Period depends on moment of inertia
Compare portrait vs landscape orientation
Shows importance of "point mass" assumption
3. "Damped Oscillations"
Plot amplitude vs time
Fit exponential: A=A0e−bt
Discuss energy loss mechanisms
Link to Q-factor concepts
Troubleshooting Your Setup
"My g value is 10.5 m/s²!"
Check length measurement (probably too short)
Verify you're measuring to centre of mass
Ensure string doesn't stretch under load
"Data looks noisy/irregular"
Phone might be rotating (not just swinging)
Secure phone better in pouch
Check for air currents (fan/aircon)
Use longer string for slower, cleaner motion
"Period changes over time"
Normal - amplitude decay affects period
Use first 10 swings only
Or apply amplitude correction factor
Excel/Sheets Analysis Template
Column A: Time (s)
Column B: Acceleration (m/s2)
Column C: Peak marker (1 if peak, 0 otherwise)
Formulas:
Period:=COUNTIF(C:C,1)/[number of oscillations]
g value:=4*PI()^2*[Length]/[Period]^2
Uncertainty:=g*SQRT((δL/L)^2+2×δT/T^2)
Beyond Basic Measurements
Cross-Validation Methods
Use gyroscope data (angular velocity peaks)
Video analysis with Tracker software
Light gate at equilibrium position
Compare all three for systematic error check
Research Extensions
Compare measured g at different locations (results should agree within your measurement uncertainty)
Test Einstein's equivalence principle
Build Foucault pendulum (Earth's rotation)
Couple two pendulums (normal modes)
Summary: Your Exam-Ready Checklist
✓ Smartphone > stopwatch for timing precision ✓ Keep amplitude < 10° to use simple formula ✓ Measure length to centre of mass carefully ✓ Plot T2 vs L for best g value ✓ First 10 swings minimize damping effects ✓ Full error propagation scores method marks ✓ Explain systematic corrections in discussion
Master this experiment and you'll handle any pendulum variation the examiner throws at you - plus you'll genuinely understand why g matters beyond just memorizing 9.81m⋅s2.
Appendix: Video Demos
Timing the period of a single pendulum for small-angle oscillations
