Pendulum Lab
Adjust length, mass, angle, gravity, and damping to explore pendulum motion. Collect data across multiple trials, plot T² vs L, and derive gravitational acceleration from the slope of the regression line.
Guided Experiment: Deriving g from T² vs L
If you increase pendulum length, how do you predict the period will change? What does the slope of T² vs L tell you?
Write your hypothesis in the Lab Report panel, then click Next.
Controls
Results
T² vs Length (Slope = 4π²/g)
Record data points at different lengths to plot T² vs L. The slope of the regression line equals 4π²/g, allowing you to derive gravitational acceleration experimentally.
Data Table
(0 rows)| # | Trial | Length (L)(m) | Period (T)(s) | T²(s²) | Derived g(m/s²) |
|---|
Reference Guide
Simple Pendulum
A simple pendulum consists of a mass (bob) suspended by a massless, inextensible rod from a fixed pivot. It swings back and forth under gravity.
Key variables are length L (pivot to bob center), angle theta from vertical, and local gravity g. The mass of the bob does not affect the period.
The exponential factor accounts for energy dissipation (damping coefficient gamma). When gamma = 0 the oscillation continues indefinitely.
Period Formula
For small angles (below about 15°), the exact nonlinear equation simplifies to a harmonic oscillator. The period is:
Period increases with length (longer rods swing slower) and decreases with gravity (stronger gravity speeds up the swing). Mass has no effect.
On the Moon (g = 1.62 m/s²) a 1 m pendulum has a period of about 4.94 s, compared to 2.01 s on Earth.
Deriving Gravity
Rearranging the period formula gives g as a function of measured T and known L:
Plotting T² vs L gives a straight line through the origin. The slope equals 4π²/g:
On Earth the theoretical slope is 4π²/9.81 ≈ 4.027 s²/m. Record at least 6 trials at different lengths for a reliable regression.
Large Angle Approximation
The small-angle formula breaks down for angles above about 15-20°. The first-order correction is:
where theta0 is in radians. At 30° the correction is about +1.7%; at 45° it is about +4%. The error is always positive - large-angle pendulums are slower than the small-angle formula predicts.
For precise gravity measurements keep the initial angle below 10° so the correction stays under 0.2%.
