Simple pendulum: T² = f(L)
Discover that a pendulum's period depends on its length but not on amplitude, and verify T = 2π√(L/g) with the FizziQ Web Pendulum simulation.
Activity Summary
The student uses the FizziQ Web Pendulum simulation to measure the oscillation period by varying the string length, then the initial angle. They verify that T is independent of amplitude (isochronism) and proportional to √L, and determine g from the slope of T² versus L.
Introduction
According to legend, Galileo discovered the isochronism of the pendulum by watching a chandelier swing in the Cathedral of Pisa. He noticed that regardless of the amplitude, the oscillations took the same time. This remarkable property made the pendulum the basis of accurate clocks for three centuries. But is the period truly independent of amplitude? And exactly how does it depend on the string length? The FizziQ Web Pendulum simulation lets you test these questions with precision, varying one parameter at a time and measuring the period from the angle-time graph.
Learning Objectives
- Verify the isochronism of small oscillations (T independent of amplitude)
- Verify T = 2π√(L/g) by plotting T² versus L
- Determine g from the slope of the T²(L) graph
- Identify the limit of the small-angle approximation
Scientific Concepts
Instruments and sensors
Scientific instruments
- FizziQ Web Pendulum simulation
FizziQ Features
- Experiment notebook — used to read the angle-versus-time recording and measure the period T between two passages at the same angle in the same direction
Required Materials
- - Computer, tablet, or smartphone with FizziQ Web - Note: the protocol relies on the FizziQ Web Pendulum simulation and remains adaptable to any equivalent simulation or video-tracking setup for angle-versus-time measurements.
Experimental Protocol
Open the Pendulum simulation in FizziQ Web (Experiment → Simulations → Pendulum).
Part 1 — Effect of length: set the initial angle to 10° (small angle). Set the length to 0.25 m.
Select angle versus time recording. Start the simulation with REC. Record a few oscillations, then stop.
In the experiment notebook, measure the period T (time between two passages at the same angle in the same direction). Note the value.
Repeat for lengths 0.50 m, 0.75 m, 1.00 m, 1.50 m, and 2.00 m. Note T for each length.
Create a table: Length L (m), Period T (s), T² (s²). Plot T² versus L. Is it a straight line through the origin?
The slope of T²(L) equals 4π²/g. Calculate g and compare with 9.81 m/s².
Part 2 — Effect of amplitude: set the length to 1.00 m. Measure the period for initial angles of 5°, 10°, 15°, 20°, 30°, 45°, and 60°.
Note T for each angle. For which angles is the period essentially identical? At what angle does the period start to increase noticeably?
Conclusion: T = 2π√(L/g) is an excellent approximation for small angles (< 20°). For large angles, the period increases slightly and depends on amplitude.
Expected Results
Part 1: T increases with length. For θ = 10°: T ≈ 1.00 s (L = 0.25 m), 1.42 s (0.50 m), 1.74 s (0.75 m), 2.01 s (1.00 m), 2.46 s (1.50 m), 2.84 s (2.00 m). The T²(L) graph is a straight line through the origin with slope 4π²/g ≈ 4.03 s²/m. Part 2: for angles up to 20°, T is essentially constant. At 45°, T increases by about 4%; at 60°, by about 7%.
Scientific Questions
- Why does the pendulum mass not appear in the period formula?
- At what angle does the small-angle approximation start to fail?
- How could you use a pendulum to measure the value of g?
- Why is the graph T²(L) more useful than T(L)?
- What would the period be on the Moon (g = 1.6 m/s²)?
Scientific Background
The **simple pendulum** is an idealized model: a point mass at the end of a massless, inextensible string, oscillating without friction. The restoring force is the tangential component of gravity: **F = -mg sin θ**.
For **small angles** (θ < 20°), sin θ ≈ θ (in radians). The equation of motion becomes **θ̈ = -(g/L) × θ**, identical to that of a harmonic oscillator with period **T = 2π√(L/g)**.
This independence of the period from amplitude is called **isochronism** of small oscillations. It is this property that made the pendulum the basis of accurate clocks from Huygens (1656) until the quartz era.
For **large angles**, sin θ ≠ θ and the approximation is no longer valid. The period increases with amplitude. The simulation lets you explore this limit directly.
The formula **T = 2π√(L/g)** shows that the period does not depend on the mass of the pendulum. It depends only on the length L and the local gravitational acceleration g.
By plotting **T² versus L**, one linearizes the relationship: T² = (4π²/g) × L. The slope of the line directly gives the value of g.
Extensions
- Plot the centripetal acceleration versus time and identify when the velocity is maximum
- Investigate the effect of very large angles (90° or more) on the motion
- Compare the simulation with a real pendulum experiment using the smartphone accelerometer
- Determine the pendulum length needed for a period of exactly 1 second
Frequently Asked Questions
How do I measure the period precisely on the graph?
Measure the time for 5 or 10 complete oscillations and divide by the number of oscillations. This is more precise than measuring a single oscillation.
The period at 30° seems the same as at 10°. Is isochronism exact?
The deviation is only about 1.7% at 30°, which may be within your measurement precision. At 60° the deviation reaches 7%, which is more clearly detectable.
Why does the period increase at large angles?
Because sin θ < θ for large angles, the restoring force is weaker than what the linear approximation predicts. The pendulum swings more slowly near the extreme positions.
Can I use this to measure g accurately?
Yes. The T²(L) graph gives g with good precision. Using 6 or more data points and a linear fit reduces measurement errors significantly.
Detailed Description
The student uses the FizziQ Web Pendulum simulation to measure the oscillation period by varying the string length, then the initial angle. They verify that T is independent of amplitude (isochronism) and proportional to √L, and determine g from the slope of T² versus L. FizziQ Web's built-in experiment notebook lets students read the recorded angle-versus-time graph and time each oscillation period directly, without any additional instrument.
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