Measuring g Using a Smartphone Pendulum for A-Level Physics Practicals

TL;DR
Ditch the stopwatch - your smartphone's accelerometer can time pendulum swings to 0.001s precision. This guide shows how to measure \(g\) to within 1% of 9.81 m/s², handle systematic errors from amplitude effects, and extract clean data using free sensor apps. Perfect for H2 Physics practical prep or home revision.

Why Your Phone Beats a Stopwatch

Traditional pendulum experiments suffer from human reaction time (~0.2s), making it hard to get \(g\) closer than ±5%. But your phone's accelerometer samples at 100-400 Hz, detecting the exact moment of direction change at each swing endpoint.

Key advantages:

• Precision timing: 0.001s resolution vs 0.01s on 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

1. Download a sensor app:
   • Android: "Physics Toolbox Sensor Suite" or "Phyphox"
   • iOS: "Sensor Logger" or "SensorLog"
2. Secure the phone:
   • Place in small pouch/sock
   • Ensure phone can't slip during swings
   • Keep centre of mass aligned with string attachment
3. Measure pendulum length \(L\):
   • From pivot point to phone's centre of mass
   • Typical good range: 0.8-1.5m
   • Record uncertainty: ±0.002m
4. 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

1. Start sensor recording before releasing pendulum
2. Release gently (no initial velocity)
3. Record 20-30 complete oscillations
4. Keep amplitude small (under 10°)
5. 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

1. Export data to spreadsheet
2. Plot acceleration vs time
3. Count peaks over 20 oscillations
4. Calculate: \(T = \frac{\text{total time}}{20}\)

Method 2: Fourier Analysis (Advanced)

1. Use FFT function in spreadsheet
2. Identify fundamental frequency \(f\)
3. Period \(T = \frac{1}{f}\)

Calculating \(g\)

For small angles, the simple pendulum formula applies:

\[T = 2\pi\sqrt{\frac{L}{g}}\]

Rearranging for \(g\):

\[g = \frac{4\pi^2L}{T^2}\]

Error Analysis That Scores Full Marks

Systematic Errors

1. Finite amplitude correction:
   • Formula assumes \(\theta → 0\)
   • For angle \(\theta_0\): \(T_\text{actual} = T_\text{theory}(1 + \frac{\theta_0^2}{16})\)
   • Keep \(\theta_0 < 10°\) for <0.1% error
2. Air resistance:
   • Causes amplitude decay
   • Use first 10 swings for cleanest data
   • Or fit exponential decay and extrapolate
3. Finite mass distribution:
   • Phone isn't a point mass
   • Measure to centre of mass carefully
   • Error typically <1% if \(L > 1m\)

Random Errors

1. Length measurement: \(\delta L = ±0.002m\)
2. Period determination: \(\delta T = ±0.001s\) (with phone)
3. Propagate to \(g\): \[ \frac{\delta g}{g} = \sqrt{\left(\frac{\delta L}{L}\right)^2 + \left(\frac{2\delta T}{T}\right)^2} \]

Graphical Analysis Methods

The \(T^2\) vs \(L\) Plot

Instead of calculating \(g\) from single measurements:

1. Vary pendulum length (0.5m to 1.5m in 0.1m steps)
2. Measure period for each length
3. Plot \(T^2\) against \(L\)
4. Gradient = \(\frac{4\pi^2}{g}\)
5. Calculate: \(g = \frac{4\pi^2}{\text{gradient}}\)

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 \(\theta_0^2\)
• 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: \(A = A_0e^{-bt}\)
• 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 \(\pu{(s)}\)
• Column B: Acceleration \(\pu{(m/s²)}\)
• Column C: Peak marker (1 if peak, 0 otherwise)

Formulas:

• Period: =COUNTIF(C:C,1)/[number of oscillations]
• g value: =4PI()^2[Length]/[Period]^2
• Uncertainty: =gSQRT((δL/L)^2+(2δT/T)^2)

Beyond Basic Measurements

Cross-Validation Methods

1. Use gyroscope data (angular velocity peaks)
2. Video analysis with Tracker software
3. Light gate at equilibrium position
4. Compare all three for systematic error check

Research Extensions

• Map \(g\) variations around Singapore \(\pu{(\pm 0.001 m/s²)}\)
• 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 \(T^2\) 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 \(\pu{9.81 m.s2}\).