# Foucault Pendulum: Deriving Equations of Motion

• ian2012
In summary, the author provides a derivation of the equations of motion for a Foucault pendulum. The equations of motion come from the definition of the Coriolis force and the equation of motion for a pendulum is: - r'' = -(g/l) r - 2ωcosθk Λ r'
ian2012
Consider a pendulum which is free to move in any direction and is sufficiently long and heavy that it will swing freely for several hours. Ignoring the vertical component both of the pendulum's motion and of the Coriolis force, the equations of motion for the bob are:

$$\ddot{x}=-\frac{g}{l}x+2(\omega)cos\theta\dot{y}$$
$$\ddot{y}=-\frac{g}{l}y-2(\omega)cos\theta\dot{x}$$

I've found these equations from 'Classical Mechanics - Kibble & Berkshire, 5th Edition'. I don't understand how they are derived?

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For me the thing that gives me difficulty in following Foucault pendulum derivations is that the author usually jumps from notation to notation. I see authors switching between index notation, vector notation and parametric notation.

I suggest you comb the internet and textbooks that you can get hold of for derivations, and piece together a picture that you comprehend.

There is a http://www.cleonis.nl/physics/phys256/foucault_pendulum.php" on my website, and in all there are three Foucault related simulations.

The applets feature true simulations, not animations.
- An animation depicts the mathematics of the analytic solution to the equation of motion.
- A simulation takes as input the raw differential equation that relates acceleration to the force(s) that act(s), and then performs numerical analysis to obtain a trajectory.

Cleonis
http://www.cleonis.nl

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Thanks Cleonis.

## 1. What is a Foucault Pendulum?

A Foucault Pendulum is a device used to demonstrate the rotation of the Earth. It consists of a long pendulum with a heavy weight at the end that swings back and forth, appearing to change direction over time due to the Earth's rotation.

## 2. How does a Foucault Pendulum work?

The motion of a Foucault Pendulum is due to the Coriolis effect, which is caused by the Earth's rotation. As the pendulum swings back and forth, the Earth rotates underneath it, causing the pendulum's path to appear to curve.

## 3. What are the equations of motion for a Foucault Pendulum?

The equations of motion for a Foucault Pendulum can be derived using the principles of classical mechanics. These equations take into account the length of the pendulum, the mass of the weight, and the Earth's rotation rate.

## 4. What factors affect the motion of a Foucault Pendulum?

The motion of a Foucault Pendulum is affected by several factors, including the length of the pendulum, the mass of the weight, the Earth's rotation rate, and the location of the pendulum on Earth. The closer the pendulum is to the Earth's poles, the more apparent the rotation will be.

## 5. What is the significance of the Foucault Pendulum in science?

The Foucault Pendulum is significant because it was the first experimental proof of the Earth's rotation. It also helped to confirm the Coriolis effect, which is a fundamental concept in meteorology and oceanography. Additionally, the equations of motion for the Foucault Pendulum have been used to understand other phenomena in physics, such as the motion of celestial bodies.

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