How Does the Foucault Pendulum Prove Earth's Rotation?

Foucault pendulum proves Earth's rotation
Can a pendulum that swings back and forth right here on the surface of Earth demonstrate and prove conclusively that the Earth rotates about its axis? Read this Buzzle post to find the answer.
Did You Know?
The Principia housed at the Oregon Convention Center is the world's largest Foucault pendulum. It comprises a 900-pound ball which is suspended using a 70-foot long cable. It swings continuously at a height of 20 feet over the heads of the visitors.
Without ogling at the heavens through a giant telescope, or making a quick trip to the International Space Station, will you be able to prove conclusively that the Earth rotates around its axis? Seems to be a near impossible task, doesn't it? However, as was demonstrated by the French physicist Jean Bernard Leon Foucault back in the 1800s, it is possible to do that, right here on Earth itself.

Born in 1819, Foucault was the son of a French publisher. From an early age he showed exceptional skill at making mechanical toys. He studied medicine, but later moved on to the study of physical sciences at the Paris Observatory. Foucault is widely regarded as one of the most gifted and versatile experimentalists of all time. He is credited with the invention of the gyroscope, which is an important instrument used to indicate a fixed direction during rapid motions. It forms an important part in many navigational systems used today.

In 1851, Foucault devised an innovative experiment to verify the Earth's rotation. His main apparatus was a large pendulum, which later came to be known as the Foucault pendulum in his honor. Here, we shall find out how the Foucault pendulum proves that the earth rotates. However, before that, let's examine a few important concepts in physics, that form the basis of this experiment.
Pendulums and the Concept of Inertia
Sir Isaac Newton explained that if an external force is applied to a body, it will begin moving in a straight line. This body will then continue its linear motion until another external force is applied to interrupt/redirect it. Newton termed this ability of a body to resist any changes in its state of motion, including both speed as well as direction of motion, as Inertia.

A good example of the effect of inertia is when the brakes in the vehicle you are sitting are applied suddenly. While the vehicle comes to a stop due to friction, inertia causes your body to continue moving forward until you stop yourself using your hands or with the help of seat belts. This is experienced as a sudden jerk.

Pendulums are devices that work on the concept of inertia. They typically comprise a bob which is usually a weighted sphere that is suspended from one end of a taut wire or string, while the other end is held in one's hand or tied to a fixed point at some height.

When the bob is pulled back and released, it descends down and moves forward under the influence of gravity and inertia receptively, until the length of the string limits its travel. After that, it swings back out and up again due to the same forces.

Thus, the pendulum continues to swing freely under the influence of gravity and inertia opposed only by the air surrounding it. The amount of inertia developed depends on the weight of the bob, which also determines the time it takes for air resistance to stop the pendulum from swinging.

The more the weight of the bob and longer the length of its swing, the more will be the time for which it will continue swinging. Air resistance will continue to counter its motion and decrease its speed, finally bringing it to a stop.
Demonstration of the Foucault's Pendulum Experiment
To understand the working of the Foucault's pendulum, lets carry out a simpler version of his original experiment. The following image displays the setup.
Foucault Pendulum With Two Fallen Pegs
Foucault pendulum with two fallen pegs
As seen in the figure above, a pendulum is constructed by suspending a sufficiently heavy ball on a long wire/string which is attached to a fixed point on the roof of a high ceiling, wide hall or room. The ball needs to be heavy and the wire long enough so that the pendulum is able to overcome the air resistance and continue swinging for a long duration.

A number of light wooden pegs are arranged circularly and equidistant from each other on the floor below the pendulum. These pegs are chosen such that they fall easily on coming in contact with the heavy ball, without affecting the velocity or altering the direction of its swing. The ball is then pulled up and tied to a stable stationary point in the room with the help of a small piece of string.

To begin the experiment, this string is burnt which frees the pendulum and sets it swinging. The reason why this method of burning a string is used, is that prevents the accidental introduction of any angular momentum, because for our experiment to work properly, we need the ball to swing back and forth in a nearly straight line.

Arranged correctly, on the very first swing of the ball, it knocks down two diagonally opposite pegs (as shown), that lay in the path of its swing. The ball then continues to oscillate without disturbing any of the other standing pegs. This is allowed to happen for a while, ensuring that the pendulum's swing and the rest of the experimental setup remains undisturbed.
Foucault Pendulum With Four Fallen Pegs
Foucault pendulum with four fallen pegs
After a sufficiently long period of time (typically about an hour or more), the pendulum knocks down another two diagonally opposite pair of pegs as shown in the image above. As more and more time elapses, the pendulum periodically keeps knocking down the standing pegs, until the air resistance finally slows it down, finally bringing it to a halt, thus ending the experiment.
Observation and Explanation
In the above demonstration, we observed that the pendulum continued to swing along the same line, under the influence of gravity and its own inertia alone. There was no external force acting on it which could have altered its trajectory. Despite that, it seemingly rotated in the plane of its swing, and was able to knock down more sets of diagonally opposite pegs.

The only explanation for this phenomenon is that while the pendulum swung back and forth, undisturbed, along a straight line, the Earth and therefore the floor beneath the pendulum moved, thus rotating the circle of pegs along with it. This caused more of the standing pegs to come in the line of the pendulum's swing, and so they too were knocked down. Essentially, the plane of oscillation of the pendulum remained constant, while the Earth beneath it moved, changing the orientation of the floor with respect to it.

Now, the way the floor below moves with respect the constant swing-plane of the pendulum, is dependent on the location on Earth's surface where this experiment is carried out. This is shown in the image below.
Respective Motion of the Floor at Different Locations
Respective motion of floor at different locations
If the experiment is carried out exactly at either the North or the South pole, the Earth and building floor will rotate beneath the pendulum's plane taking one sidereal day (23.93 hours) to complete the rotation and make all the pegs fall.
If the experiment is conducted somewhere along the Equator, the building will co-rotate along with the Earth, traveling eastward along the Earth's axis. As there will be no twisting of the floor, after the initial pair of pegs fall, none of the other pegs will be knocked off.
If the experiment is conducted at some place between the pole and the Equator, the motion of the building floor, viewed with respect to the constant swing-plane of the pendulum, will be a combination of rotation and traveling. This will result in it being twisted beneath the pendulum, albeit at a slower rate than the rotation of the Earth. The pendulum doesn't share this twisting motion, and hence, its plane of swing remains constant while the floor changes orientation so that the pegs are knocked out sequentially.
The degree of twist per day (n) is given by the following formula:

N = 360º sin φ

If the Foucault's pendulum is used at 30º south latitude, it will twist by 180º per day and will complete a full rotation in two days.
Thus, with the help of a Foucault pendulum, we can conclusively prove that the Earth rotates about its axis.

The Original Foucault's Pendulum Experiment

To prove that the Earth rotates, Leon Foucault carried out the pendulum experiment at the Meridian of the Paris Observatory in February 1851. Some weeks later, he carried out the demonstration once more with the help of a 28 kg, brass-coated lead bob which was suspended from the dome of the Pantheon in Paris, with the help of a 67-meter long metal wire.

Once this pendulum began oscillating, it was observed that the pendulum swings regularly changed directions in the clockwise direction at the rate of approximately 11 degrees per hour completing a full circular rotation in 32.7 hours, thus proving that the Earth rotates.
In conclusion, the Foucault pendulum is one the simplest and most elegant ways to prove Earth's rotation. Since its initial demonstration, it has garnered a lot of interest in the general as well as the scientific communities. Today, there are several Foucault pendulums around the world, found mainly at universities, science museums, and planetariums. To prevent air resistance from slowing down and stopping the pendulums, many of these places utilize electromagnetic drives, motors, etc., to keep the bob oscillating continuously for the audience to see. Thus, the Foucault pendulum experiment still finds a lot of appeal among the young and old alike.