In 1851, French physicist Léon Foucault suspended a massive iron bob from the dome of the Panthéon in Paris. As the pendulum swung freely, it slowly changed direction over the course of the day. Visitors were mesmerized because the pendulum appeared to rotate all by itself. In reality, it was the Earth turning beneath the oscillating bob. This elegant demonstration became one of the most direct proofs that Earth rotates, providing an observable phenomenon that anyone could witness without astronomical equipment.
The rate at which the pendulum's plane of oscillation rotates depends on the latitude of the experiment. At the North or South Pole, the swing plane completes a full circle every sidereal day. At the equator, however, there is no precession at all. Everywhere in between, the rotation rate is proportional to the sine of the latitude. Mathematically, the precession angular speed is expressed as where is Earth's rotation rate of about 15 degrees per hour.
Our calculator implements this formula to show how quickly a Foucault pendulum would appear to precess at any given latitude. Enter your latitude, and the script displays two useful outputs: the rotation in degrees per day and the time required for the swing plane to complete a full 360-degree turn. These values provide insight into how the effect becomes more pronounced as you move toward the poles.
Why does the pendulum precess? From the perspective of an inertial frame in space, the pendulum's plane remains fixed. However, because Earth rotates underneath it, a stationary observer on the ground perceives the pendulum's orientation to drift over time. The component of Earth's angular velocity that is perpendicular to the pendulum's axis of rotation is what causes the precession. At the equator, this component is zero, while at the poles it equals Earth's full rotation rate.
The length of the pendulum determines its oscillation period according to . Although the period does not affect the precession rate, it does influence how stable the motion appears. Longer pendulums swing more slowly, so the gradual change in direction is easier to observe. Many museums with Foucault pendulum exhibits use wires tens of meters long to achieve a long period and minimize frictional effects.
Foucault's public demonstration quickly became famous, spurring similar pendulum installations around the world. The experiment required no telescope or complicated apparatus, so it resonated with a broad audience at a time when the rotation of Earth was still a source of philosophical and religious debate. Today, Foucault pendulums remain popular educational tools in science museums, universities, and even some churches, illustrating the interplay between rotational motion and gravity with mesmerizing simplicity.
The full derivation of the precession formula uses the Coriolis force in a rotating reference frame. When the pendulum swings north-south, the Coriolis force pushes it sideways, gradually twisting the plane of motion. By analyzing the equations of motion, one finds that the precession angular frequency is . Integrating over time shows that after one day, the plane has rotated by . The closer the pendulum is to either pole, the greater the daily rotation.
Foucault pendulums are not perfect: air resistance and mechanical friction slowly damp the motion, so real exhibits often use a small drive mechanism to maintain amplitude. Additionally, local vibrations can disturb the swing. Yet despite these challenges, many large installations display the precession beautifully over several hours. Watching the bob knock over a ring of markers hour by hour provides a tangible connection to the planet's rotation that otherwise goes unnoticed in our daily lives.
In the classroom, teachers use small-scale versions or computer simulations to explore rotational dynamics. By adjusting the latitude in our calculator, students can compare what would happen in different cities around the world. This encourages discussion of how Earth's rotation rate remains constant while the observed precession depends solely on latitude. Connecting the experiment to concepts like inertial frames and the Coriolis effect deepens students' appreciation for classical mechanics.
The Foucault pendulum is more than just a clever physics demonstration; it is a vivid reminder of our planet's place in space. From satellite launches that must account for Earth's spin to the swirling patterns of hurricanes, rotation shapes countless phenomena. Exploring the pendulum's precession with this calculator helps bridge the gap between everyday experience and global, even cosmic, scales.
By entering a latitude into this tool, you can immediately see how the Foucault pendulum reveals Earth's rotation. The simple dependence on means that the effect is strongest near the poles and vanishes at the equator. Use the results to plan a demonstration, interpret a museum exhibit, or satisfy your curiosity about a classic experiment that elegantly ties together geometry, physics, and our planet's ceaseless spin.
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