Mr. Scandrett's ClassroomOS

Physics · Lesson 13

The Pendulum

Periodic Motion Potential ↔ Kinetic Energy Simple Harmonic Motion

A weight on a string, swinging back and forth under gravity. The pendulum is one of the oldest mechanical timekeepers and one of the most elegant demonstrations of energy conservation — potential and kinetic energy trading back and forth with each swing.

Galileo and the Pendulum

Portrait of Galileo Galilei

Galileo Galilei (1564–1642) reportedly observed a chandelier swinging in the Cathedral of Pisa and noticed something remarkable: regardless of how wide the swing, it seemed to take the same time to complete each oscillation. He tested this using his own pulse as a timer. This property — constant period for small angles — led to the development of pendulum clocks, which were the world's most accurate timekeepers for nearly 300 years.

The Period Formula

For small angles, the period T of a simple pendulum depends only on its length and gravitational acceleration — not on the mass of the bob or the amplitude:

Key insight: Mass does not appear in the formula. A heavier bob swings at exactly the same rate as a lighter one of the same length. This surprised many people in Galileo's time.

Energy Exchange

At the highest points of the swing, the bob momentarily stops — all energy is gravitational potential energy (PE = mgh). At the lowest point (the center), the bob moves fastest — all energy is kinetic (KE = ½mv²). Throughout the swing, the total energy is conserved: PE + KE = constant.

Energy Conservation Lab

Drag the bob to set the release angle, then watch potential and kinetic energy trade places in real time. Switch environments, add air friction, and turn on the strobe trail to make the motion easier to read.

Angle0°
Speed0.00 m/s
EnvironmentEarth
Total Energy0.00 J

Without friction, PE + KE stays nearly constant. Add drag and the missing mechanical energy reappears in the orange heat bar.

Real-World Applications

🕰️
Pendulum ClocksInvented by Christiaan Huygens in 1656. Each swing of a calibrated pendulum moves the clock mechanism by one precise increment.
🌍
Measuring gScientists measured gravitational acceleration at different latitudes by timing pendulums. Faster period = stronger gravity = slightly flattened Earth.
🏗️
Tuned Mass DampersSkyscrapers like Taipei 101 use giant suspended pendulums to counteract wind oscillations and prevent swaying in typhoons.
🔬
SeismometersPendulums detect ground motion. When the Earth shakes, the bob lags behind the frame, recording the motion of earthquakes.

Practice Problems

Use T = 2π√(L/g). g = 9.81 m/s². Round to 2 decimal places.

Easy1. A pendulum has length 1 m on Earth. What is its period? (2π√(1/9.81))

Hint: T = 2π√(1/9.81) = 2π × 0.319 ≈ 2.01 s

Easy2. If you double the length of a pendulum, what happens to its period?

Hint: T ∝ √L. Double L → √(2L) = √2 × √L → period × √2.

Medium3. A pendulum on the Moon (g = 1.62 m/s²) has length 0.5 m. What is its period?

Hint: T = 2π√(0.5/1.62) = 2π√0.309 ≈ 3.49 s

Challenge4. A grandfather clock pendulum has a period of exactly 2 s. What must its length be? (L = g(T/2π)²)

Hint: L = 9.81 × (2/2π)² = 9.81 × 0.1013 ≈ 0.993 m