Partnership for Integration of Computation into Undergraduate Physics

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 $\Omega$.