All Labs

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

Length1.00 m
Mass0.10 kg
Initial Angle15.00 °
Gravity9.81 m/s²
Damping0.00

Results

T=2π1.009.81=2.0061 sT = 2\pi\sqrt{\dfrac{1.00}{9.81}} = 2.0061\text{ s}
Period (small angle)
2.0061 s
Frequency
0.4985 Hz
Period (large angle)
2.0147 s
Large angle correction
+0.428%
Derived g
9.8100 m/s²
Max velocity
0.820 m/s
Current angle
15.000°
g=4π2LT2=4π2×1.002.00612≈9.8100 m/s2g = \dfrac{4\pi^2 L}{T^2} = \dfrac{4\pi^2 \times 1.00}{2.0061^2} \approx 9.8100\text{ m/s}^2

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)
#TrialLength (L)(m)Period (T)(s)T²(s²)Derived g(m/s²)
0 / 500
0 / 500
0 / 500

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.

θ(t)=θ0cos⁡ ⁣(2πtT)e−γt\theta(t) = \theta_0 \cos\!\left(\frac{2\pi t}{T}\right) e^{-\gamma t}

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:

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

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:

g=4π2LT2g = \frac{4\pi^2 L}{T^2}

Plotting T² vs L gives a straight line through the origin. The slope equals 4π²/g:

T2=4π2g⋅L⇒slope=4π2gT^2 = \frac{4\pi^2}{g}\cdot L \quad\Rightarrow\quad \text{slope} = \frac{4\pi^2}{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:

T≈T0(1+θ0216)T \approx T_0\left(1 + \frac{\theta_0^2}{16}\right)

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%.