Theory — The Simple Pendulum
What a Pendulum Is
A simple pendulum is a small heavy bob hanging from a light string of length L, free to swing back and forth. When pulled aside and released, gravity pulls it back toward the lowest point, it overshoots, and it swings to the other side — repeating in a regular rhythm. One complete back-and-forth swing is one period T, measured in seconds.
The Period Equation
For small swings, the period of a simple pendulum depends on only two things: the length of the string and the local acceleration due to gravity. Remarkably, it does not depend on the mass of the bob or (for small angles) on how far you pull it aside.
T = period (s), L = length (m), g = 9.8 m/s² on Earth
Why Mass and Amplitude Don't Matter
Mass cancels out: a heavier bob is pulled harder by gravity, but it also has more inertia resisting that pull, and the two effects exactly cancel — just as all objects fall at the same rate. Amplitude doesn't matter (for small swings) because a wider swing covers more distance but the restoring force is correspondingly larger, so the bob moves faster and the time stays the same. This property — a period independent of amplitude — is called isochronism and is why pendulums make good clocks.
Period Squared and the Straight-Line Graph
Squaring the period equation turns the square-root relationship into a straight line, which is far easier to analyze. A graph of T² against L is a straight line through the origin whose slope is 4π²/g — so measuring the slope lets you calculate g.
slope of T² vs L = 4π² / g → g = 4π² / slope
Increase the length
Longer string → longer period. Because T ∝ √L, quadrupling the length doubles the period.
Change mass or amplitude
No effect on the period (for small swings). The period depends only on L and g.
| Change | Effect on period T | Reason |
|---|---|---|
| Increase length L | increases (T ∝ √L) | T = 2π√(L/g) |
| Increase bob mass | no change | mass cancels (like free fall) |
| Increase amplitude (small) | no change | isochronism |
| Weaker gravity (e.g. Moon) | increases (T ∝ 1/√g) | T = 2π√(L/g) |
Instructions — Running the Virtual Experiment
The Pendulum tab lets you swing the bob and time its period; the Find g tab records period versus length so you can measure gravity. Record every reading in your lab notebook.
Simulation — Timing the Pendulum & Measuring g
Controls
Set length (Earth gravity)
| Length L (m) | Period T (s) | T² (s²) | T²/L (s²/m) |
|---|---|---|---|
| No readings yet — pick a length and click "Record reading". | |||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission.
Pendulum Lab: Period, Length, and the Measurement of g
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To investigate how the period of a simple pendulum depends on its length, mass, and amplitude, to verify the relationship T = 2π√(L/g), and to determine the local acceleration due to gravity from the slope of a graph of period-squared versus length.
Theory
For small swings the period of a simple pendulum depends only on its length and the acceleration due to gravity, not on the mass of the bob or the amplitude. Squaring the period relation gives a linear equation whose slope yields g.
T² = (4π²/g)·L → slope = 4π²/g
g = 4π² / slope
Calculations — Sample: L = 1.00 m on Earth
Predicted period: T = 2π√(1.00/9.8) = 2π(0.3194) = 2.01 s
Period squared: T² = (2.01)² = 4.03 s²; T²/L = 4.03/1.00 = 4.03 s²/m
g from the slope: slope = 4.03 s²/m → g = 4π²/4.03 = 39.48/4.03 = 9.80 m/s²
Percent error: |9.80 − 9.8|/9.8 × 100% = 0.0%
Results Table — Period vs Length (Earth, small amplitude)
| Length L (m) | Period T (s) | T² (s²) | T²/L (s²/m) |
|---|---|---|---|
| 0.25 | 1.00 | 1.01 | 4.03 |
| 0.50 | 1.42 | 2.01 | 4.03 |
| 1.00 | 2.01 | 4.03 | 4.03 |
| 1.50 | 2.46 | 6.04 | 4.03 |
| 2.00 | 2.84 | 8.06 | 4.03 |
Control trials: changing the bob mass from 1 kg to 5 kg left the period unchanged; changing the amplitude from 5° to 30° changed the period by less than 2%. Both confirm the period's independence from mass and (small) amplitude.
Discussion
The measured periods matched the prediction T = 2π√(L/g) at every length. The quantity T²/L was constant at 4.03 s²/m across the whole range, confirming that T² is proportional to L. A graph of T² against L was a straight line through the origin with slope 4.03 s²/m. Using g = 4π²/slope gave g = 9.80 m/s², in excellent agreement with the accepted value of 9.8 m/s².
The control experiments confirmed the two surprising predictions of the theory. Changing the mass of the bob had no measurable effect on the period, because gravitational pull and inertia scale together and cancel. Changing the amplitude (for small angles) also left the period essentially unchanged, the property of isochronism that makes pendulums useful as timekeepers. Reducing gravity (the Moon setting) lengthened the period, consistent with T ∝ 1/√g.
Conclusion
The experiment verified that the period of a simple pendulum is given by T = 2π√(L/g), depending only on length and gravity. The period was independent of mass and of small amplitude. From the slope of the T²-versus-L graph the acceleration due to gravity was found to be 9.80 m/s², matching the accepted value and confirming the pendulum as an accurate method for measuring g.
Practice Questions
Show all work and include units in your answers.