Introductory Pendulum Lab: Determining g and Examining Energy Conservation

Pendulum Motion Basics: Period, Length, and Small-Angle Experiments

Overview

A simple pendulum (mass on a light string) exhibits nearly periodic motion for small displacement angles. The period depends mainly on pendulum length and gravity; mass has negligible effect. Small-angle approximation (θ ≲ 10°) simplifies analysis to simple harmonic motion.

Key equations

• Period (small-angle):

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T = 2π √(L / g) 

• Angular frequency:

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ω = 2π / T = √(g / L) 

• Restoring torque (small θ):

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τ ≈ -m g L θ 

(leading to simple harmonic motion when sinθ ≈ θ)

Typical experimental goals

• Measure T for various L to verify T ∝ √L.
• Determine local g by fitting T^2 vs L (slope = 4π^2 / g).
• Test small-angle validity by comparing periods at increasing amplitudes.
• Estimate uncertainties and propagate them to g.

Suggested procedure (classroom-ready)

1. Set up: Suspend a small dense bob from a fixed pivot with a low-mass string. Measure L from pivot to center of mass of bob.
2. Amplitude: Displace to a small angle (≈5°) for baseline measurements.
3. Timing: Release and time N oscillations (N = 10–20) using a stopwatch or photogate; repeat 3–5 trials per length.
4. Vary L: Record periods for at least 5 different lengths (e.g., 0.30–1.00 m).
5. Amplitude test: Repeat for larger angles (10°, 20°, 30°) to observe deviation from small-angle theory.
6. Data: Compute T = measured time / N; compute mean and standard uncertainty.

Data analysis

• Plot T vs √L or T^2 vs L. Fit linear model to T^2 = (4π^2 / g) L + intercept.
• Extract g = 4π^2 / slope. Include uncertainty from fit.
• Compare periods at different amplitudes; percent difference indicates small-angle breakdown.

Sources of error & tips

• Measure effective length accurately (pivot to bob center).
• Minimize air currents and use dense compact bob to reduce air drag.
• Keep amplitudes small for SHM; if using larger angles, use full nonlinear period equation or numerical integration.
• Reduce timing error by timing many oscillations and using electronic timing if available.

Quick experimental example (reasonable defaults)

• Lengths: 0.30, 0.45, 0.60, 0.75, 0.90 m.
• N = 10 oscillations, 5 trials each, small angle 5°.
• Expect T for 0.60 m ≈ 1.55 s; fit should yield g ≈ 9.7–9.9 m/s^2 in a typical undergraduate lab with moderate errors.

If you want, I can generate a full lab handout with equipment list, step-by-step instructions, data table, and sample analysis.