Foucault’s Pendulum

Objective

The purpose of this research is to explain with detail what happened when Foucault used a pendulum to demonstrate the rotation of the Earth. The second part consists in interpreting the philosophy behind this demonstration.

Abstract

This work is divided on several sections; the first one is an historical background to describe what Foucault did and what he wanted to prove. Then I will determine the variables that are going to be used to describe the motion of the pendulum. After that comes an exhaustive analysis of the data, which includes the solution of some differential equations using Laplace transformation; calculus for the velocity, position, and acceleration; a discussion about energy and the effects of air resistance on the pendulum. The work also includes graphs and tables as support. Next, we will discuss all the philosophy behind this amazing pendulum to end with conclusions and personal interpretations.It is important to mention that some parts of the equations aren’t explained deeply because we had not enough knowledge in those sections; therefore they are assumed as a necessary step to achieve our objective. Let’s start defining basic concepts such as the pendulum before we move on to the historical background. What is a pendulum? The word pendulum comes from the Latin word “pendulus” which means “hanging”. A pendulum is something that hangs of a fixed reference and that swings back and forth due to the weight and the inertia of the body. Why pendulums are important? Pendulums can be used to provide accurate time keeping, to measure the acceleration due to gravity, and also to show that the earth spins.The Foucault pendulum was used by J. Foucault to prove experimentally the rotation of the earth. It also helps us to understand the principle of inertia of Newton, which tells that any object will remain in state of motion (with a constant velocity) or at rest unless any force is applied upon it. By the time the pendulum oscillates, it seems that the pendulum is changing its direction, however, the earth is the body that’s changing the plane of reference; it’s rotating!Let’s use an example to explain better these phenomena. For a person placed on the Earth, they will see as if the pendulum was rotating; however if an alien located in the outer space saw the pendulum, he would see that the earth is rotating but the pendulum stays the same. The pendulum is a frame of reference for every point in the space, even for the motion of the universe.

Historical Background (The experiment)

The Foucault’s pendulum is free to oscillate in a vertical plane, the direction of the pendulum stays always the same but it seems to rotate, why? The answer is simple: the Earth is moving in its axis, so she’s the one who is changing of direction. We can think that the pendulum is a fixed reference, not only for the rotation of the earth, but also for the universe.

The first public exposition occurred in 1851 in the Observatory of Paris. Nonetheless, the pendulum became more famous when Foucault suspended a 28 kg spherical mass with a 67 m wire from the Pantheon vault in Paris. The plane of oscillation rotated 11o degrees to the right per hour completing a circle in 32.7 hours. This was the first dynamic prove of the rotation of the earth. One year later Foucault invented the Gyroscope, a flywheel device that is pivoted freely about its center of gravity. In rapid motion its axis indicates fixed direction in space.

Why did he choose a long wire? The reason it’s very simple: the longer the wire of a Pendulum, the longer and slower are the oscillations. This is due to the Simple Harmonic Motion. If Foucault would have chosen a smaller pendulum he might have to increase the angle that gave the amplitude and therefore the oscillations would last less.

The plane of oscillation will change depending the location of the pendulum, for the North Pole the plane will rotate in one direction (to the right), while in the South Pole it will be the opposite one (to the left). This can be compared with the Coriolis effect.

A Foucault pendulum requires a lot of precision, the initial launch is critic; the traditional way to do this is using a flame to cut a rope, this way we prevent lateral oscillations that might affect the motion of the pendulum.The air resistance slows the motion of the pendulum; therefore we are not talking about a simple harmonic motion but of a damped motion. To keep the system moving museums use an electromagnetic impulse; this is called a driven motion but we are not going to describe it.

Which is the mechanism that keeps the pendulum moving with the same Amplitude? The pendulum is supported, at the top, in such a way that it is freely to swing in any direction. As we have demonstrated the wire may be very long because the longer the wire, the slower the ball swings and the less the friction of the air retards it. But the real “trick” is an iron collar, located almost at the top, surrounded by an electromagnet that attracts the wire as the ball swings.

The approach

Foucault’s pendulum can be studied from a dynamic point of view. The branch of dynamics that would study this type of motion is related to vibrations and oscillations.

A pendulum is a mechanical vibration, in other words a particle or body that oscillates about a position of equilibrium. A mechanical vibration generally results when a system is displaced from a position of stable equilibrium. The system tends to return to this position under the action of restoring forces (gravitational force for the pendulum).

When the motion depends only of the restoring forces the vibration is considered a free vibration. When a periodic force is applied to the system, the resulting motion is described as forced vibration. When the effects of friction can be neglected, the vibrations are said to be undamped; however all vibrations are damped. The damping forces affect directly the amplitude of the vibration.

The Foucault’s pendulum as a SHM

Let’s start from the easiest analysis and then move to complex descriptions of the Foucault’s pendulum. The easiest way to study this motion is considering it a Simple Harmonic Motion (SHM). We know that the equation of the simple harmonic motion is given by:

This is a second order linear, homogeneous ordinary differential equation. That can be solved using Laplace transformation. The reason to do this is because we don’t know differential equations and this method converts a differential equation to an algebraic equation.First of all we need to find several transformations of Laplace. We know that Laplace transformation is found by:

The procedure is to make a table for the next functions and its corresponding transformation.

For we have:

For we solved the integral by parts:

Now let’s solve for :

For sine and cosine we have to use Euler’s transformation:

If “” was a complex number “i “then we would have:

and using Euler’s transformation:

Then we can find the Laplace transformation for sine and cosine demonstrated below:

and we know from before that:

So:

Rationalizing:

Then we can define:

We are also going to need the transformation of a derivative and second derivative.

Starting with the transformation of the derivative of any function:

Solving the integral by parts.

For a transformation of a second derivative we get:

After all the hard work done we can make a table of the functions and it’s corresponding Laplace transformations.

Solving the differential equation of the Simple Harmonic Motion by Laplace:

Using trigonometric identities we get:

Where:

is the natural circular frequency defined as: R is the amplitude. t = time. is a fase constant.

What does this mean? First we need to remember several concepts of the wave (or motion):

Period is the time interval required for the system to complete a full cycle of motion.

The number of cycles per unit of time is called frequency. The maximum displacement of the system from its position of equilibrium

is called the amplitude of the vibration.

If we want to calculate the velocity in any SHM we have to derivate the function in respect to time, and the result is:

For the acceleration, we derivate twice the function of the position respect to time:

It is important to emphasize that all this procedure is just for a Simple Harmonic Motion. This type of movement is a valid approximation for angles that are equal or lower than 100.

Period T=2 /π ω

Χ

ωotR

The variables of Foucault:

Let’s consider the pendulum that Foucault used in the Observatory of Paris, but to simplify our calculus let’s translate it to the north pole, just in the axis of the Earth, then our initial data for a SHM is (neglecting air resistance): Mass = 28 kgLength = 67 mT = 24h=10o

material: brass

Calculus:

First of all we need to calculate the amplitude of our pendulum. To do this is necessary to know the radius of the ball.If we know that the “bob” is made of brass, then we have its density:

After that, we can find the total length:

With the total length and the angle we can find the Amplitude:

This means that Foucault did his experiment in the range of a SHM, how did he did that? He used a long wire and a heavy mass so that he was able to neglect the weight of the wire. In addition the amplitude was big enough to see the change with respect to the ground. Remember that for small angles sin()=0 and cos()=1.

10o

Mb=28kg

L=67m

Bob made of brass

W

The next step is to find the solution for the differential equation; this is done with a sum of Torques respect to the support of the rope.

If we compare the equation with the one of the SHM, we would find that:

Then we can find the period:

And the frequency:

Assuming that the bob is just at the value of the amplitude to find the phase constant we have:(Taking the origin in the point of equilibrium, when the wire is vertical)

xo=11.65mvo=00=0.3824 rad/s

Then for t=0s:

11.65 = 11.65 sin(0.3824t+)

1 = sin()

=/2

This phase constant is for t=0 this means when we drop the “Bob”.

Finally the formula for the Simple Harmonic Motion can be written as:

Which is the maximum velocity, and at which point is achieved?We can find the maximum velocity when the “bob” is just in the equilibrium point, when the wire is vertical.

Which is the maximum acceleration, and where does it happen?

The greatest acceleration is achieved at the maximum value of the Amplitude. When the velocity is maximum, the acceleration is zero and vice versa.

Vmaxamax

X=0X=-R X=RK=0Umax

K=0Umax

Kmax

U=0

amax

We have told that the simple harmonic motion will not describe completely the motion of the pendulum, maybe the damped approximation will be more accurate; with it we can find the time for the pendulum to stop.

The Foucault pendulum as a Damped Motion

For a damped motion we have to add a retarding force that is determined by a constant and the velocity of the system. Then we have that the differential equation for a damped motion is:

It is important to notice that if the constant given in the retarding force is zero (=0), then the equation became a SHM. In our case, the air resistance gives this constant.

How do we find the air resistance for a sphere? The air resistance of a body depends directly on the surface of contact “S”, on the form “f” and on a constant of the air “k”. Then we can write:

The force that gives the resistance is: For a sphere we have:

f=1/2kW = 0,6 N×s2/m4

S=r2

The surface in contact is:

Therefore,

And the maximum air resistance will be:

The solution for the differential equation of the damped motion is presented in the following pages, but the final answer is:

Substituting all the literals for numbers we would get:

To know the time necessary to make the pendulum stop, we need to find the velocity:

Because it is really difficult to solve for the time when v=0, we can use excel as a tool to find that critical time. The pages of excel that contain the data for the graph are located at the end of the document. Meanwhile let’s analyze the Velocity versus Time plot.

0 5000 10000 15000 20000 25000 30000

-5

-4

-3

-2

-1

0

1

2

3

4

5

Velocity VS Time

Velocity V.S Time

As we can see, the maximum velocity of the pendulum is around the 4 m/s, which is close to what we got in the SHM analysis. This pendulum is an actual example of simple harmonic motion, that’s why it takes so long to stop.

As the graph shows, around the 25000 seconds the pendulum has a very low velocity; this would be around the 7 hours. As we can see, the velocity decays exponentially. We know that the acceleration is the derivate (the slope) of the velocity; consequently, it is possible to say that the pendulum has a negative acceleration and that it decreases it’s magnitude while the time increases.

This is what really happened when Foucault drop his pendulum in the Observatory of Paris. At that time they didn’t had the mechanism that kept the pendulum going, however the experiment resulted because of the design of the system. Finally we can say that we have achieved our objective; the forced or driven motion won’t be analyzed because we just wanted to know what occurred when Foucault demonstrated the rotation of the earth.

Relation between the rotation of the Earth and Foucault’s Pendulum

The period of the pendulum (related to the time it takes to complete a circle) change as a function of the latitude of the earth at which the system is located; the equation that describe the period is the following:

From the equation presented before we can get important information.

If we are located in the equator, then the latitude of the earth is 0, and the sin(0) = 0 ,this means that any pendulum located at this point will not rotate; the period of rotation will be infinite. In fact, the plane of reference will stay the same all the time. Foucault was lucky to choose a different of the equator to place his pendulum; otherwise he might have thought that the earth doesn’t rotate.

Let’s analyze the situation for the north pole; in that point the latitude is 90, then sin(90)=1 , surprisely the period of the pendulum will be of 24 hours, or one day.

What happens in between? For our convenience let’s see what will occur in Mexico City. The latitude for Mexico city is 19o then if we use the formula for the period of oscillation we gat that T=73.72 hours.

Philosophical interpretation

As we have discussed before, Foucault founded that if he used a pendulum made of a heavy mass and a long wire, after some period of time the plane of reference below the pendulum would have changed due to the rotation of the earth.Imagine that you could see not only the earth, but the stars, the galaxy, and all the outer space. Foucault pendulum will be exactly on the same place where he began to oscillate, while all the other elements will be moving. That’s why this pendulum is told to be a reference frame for any system. In fact Foucault was so surprised with that discovery that he worked on to develop another fixed frame of reference and that’s how he created the Gyroscope.Because all the universe has entropy and the pendulum hasn’t this device can be used to measure the disorder of whatever we imagine. The pendulum measures the rotation of any system, but respect to what, respect to a whole? In philosophy is amazing to thought that there exists a device on Earth that knows so much from the universe; Is it possible to set a Foucault pendulum and relate it’s motion to the displacement of the universe? If it is, scientists could be able to figure out how does the universe is changing and not only that, it would help to find out how or where did all began. In other words this pendulum could be the tool we’ve been looking for to find where did we come from.

Conclusions

After all the procedure, we can say that the initial objective was achieved. We’ve seen that Foucault designed his pendulum in such a way that its motion could be described as a Simple Harmonic Motion. This simplifies the calculus a lot, however his system would eventually stop due to the friction of the air. To keep the oscillations more time he had to use a “bob” (heavy weight) attached to a long wire. Finally he could determine the rotation of the Earth, which resembles to the period of the rotation of the pendulum depending on the latitude at which we are located. Maybe Foucault just wanted to show that our planet has a rotational motion on a fixed axis; however his invention goes beyond that because it is able to describe the motion of the universe. Philosophy and other sciences were such amazed that they also began to study this wonderful system.If we talked about the results, we founded that the pendulum will last 7 hours to stop, that his Amplitude of 11.65m was quite long and that the maximum velocity achieved was of the order of 4.45m/s. During this work we also proved that the Simple Harmonic Motion is a valid approximation of any mechanism only if the oscillations do not exceed the 100. It is also shown that Laplace transformation may be a difficult method to solve differential equations but it is useful if we want to convert a differential equation onto an algebraic equation.Finally I can gladly tell that this work was very helpful to understand damped motion, because it never was a priority in physics course. I would like to try to solve the Driven motion equation by Laplace and then apply it to Foucault’s pendulum just to see how the museums keep the pendulum oscillating forever; but that might be long and wasn’t a priority in this research.

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