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Foucault Pendulum

Developed by Andy Rundquist - Published October 3, 2016

This set of exercises guides the student in exploring how to apply noninertial forces in a numeric setting. One of the most common problems that students work with is the Foucault Pendulum. In these exercises students will explore how to model the pendulum in both an inertial and noninertial frame.
Subject Area Mechanics Beyond the First Year Mathematica Students who complete this set of exercises will * be able to plot the approximate trajectory based on the typical textbook approach (**Exercise 1**), * determine the full equations of motion for the inertial frame, making clear what the true origin is (center of the earth) and how all the forces work (**Exercise 2**), * plot the motion of the pendulum in the inertial frame with a "camera" that moves with the earth (**Exercise 3**), * determine the full equations of motion for the noninertial frame, making clear how all the forces work (**Exercise 4**), * plot the motion of the pendulum in the noninertial frame (**Exercise 5**) * compare the two approaches (**Exercise 6**) * explore what happens when you change the oscillation speed (**Exercise 6**) 60 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

1. The wikipedia explanation that makes several approximations determines the equations of motion to be: $$\dfrac{d^2x}{dt^2} = -\omega^2 x + 2 \Omega \dfrac{dy}{dt} \sin(\varphi)$$ and $$\dfrac{d^2y}{dt^2} = -\omega^2 y - 2 \Omega \dfrac{dx}{dt} \sin(\varphi)$$ where $\varphi$ is the latitude on earth, $\Omega$ is the angular rotation of the earth, and $\omega$ is the resonant frequency of the pendulum. Further assuming that $\omega$ is much larger than $\Omega$, these have a solution that shows that the plane of oscillation rotates at a rate of $360 \sin(\varphi)$ per day. By integrating the differential equations, show that final result. 2. Determine the (non-approximate) equations of motion for the x, y, and z coordinates of the pendulum bob assuming an origin at the center of the earth. 3. Use a numeric differential equation solver to solve the equations in (2) and make an animation of the motion of the pendulum, keeping the "camera" location at the pendulum pivot pointing toward the center of the earth. A different approach to thinking about the camera is to determine the component of the motion that is "east" and "north," recognizing that both east and north change with time (as determined by the pendulum support's motion). 4. Determine the (non-approximate) equations of motion for the x, y, and z coordinates of the pendulum bob in the rotating frame of the pivot point. 5. Use a numeric differential equation solver to solve the equations in (4) and make an animation of the motion. 6. Compare and contrast the results of (1), (3), and (5) for various values of $\Omega$.