In Taylor, read sections 9.9 to 9.10 for today, and 6.1 to 6.2 for Monday.
The Foucault pendulum is a striking demonstration of the earth's rotation (or technically that the surface is a non-inertial frame) that can be seen at a numerous museums around the world. The basic idea behind the Foucault pendulum is easy to visualize if one imagines locating the pendulum at the north or south pole. The basic idea is to decouple the pendulum from the rotation of the earth by decoupling it from the rotation of the structure from which it is suspended, and to have a pendulum that will swing sufficiently long to see the effect of the earth's rotation (at least several hours, preferably more than a day. Imagine the pendulum at the south pole, set in motion, and decoupled from the earth's rotation. As seen by an observer in space, the pendulum swings always in the same plane while the earth rotates beneath it. As seen by an observer on the ground, the plane of the pendulum's swing moves (precesses), changing by 90° in 6 hours, 180° in 12 hours, 270° in 18 hours, and a full 360° in 24 hours.
There are technical details in how to make such a pendulum: it needs a massive weight hanging from the end of a long wire, possibly with a gimbled mount. We'll ignore these details and look at the motion.
Let's begin with the forces. There are two regular forces on the bob, gravity and the tension in the wire. There are two non-inertial forces, the centrifugal force and the Coriolis force. Choose a coordinate system with z pointing up, y pointing north, x pointing east, and the origin at the equilibrium location of the bob. The equation of motion is
where r is the position vector of the pendulum bob measured from its equilibrium position. As before, the local gravitational acceleration g is a combination of the acceleration due solely to Earth's mass and the acceleration of the origin of coordinates. We'll restrict our analysis to small oscillations -- this is generally the case for real Foucault pendula -- and neglect the small changes to r as the pendulum swings and if the angle of the bob from its equilibrium location is β, then sinβ ≈ β, and cosβ ≈ 1. This means that mg ≈ Tz = T cosβ ≈ T. We allow the pendulum to swing in any direction -- as you will see, this is the point of the Foucault pendulum. Therefore, the bob can be at any x-y location, and the tension has x and y components, Tx = -m g x/L and Ty = -mgy/L. The angular velocity Ω = Ω ( sinθ j + cosθ k ) and Coriolis force has x and y components of v × Ω = Ω [ ( vy cosθ - vz sinθ ) i - vx cosθ j + vx sinθ k. In the small angle approximation, we can neglect vz, yielding the x and y equations of motion
For a simple pendulum, g/L = ω0². And Ω cosθ = Ωz, the z component of the earth's angular velocity.
We can decouple these equations by introducing the complex variable η = x + i y. Then ηdot = xdot + i ydot, and ηddot = xddot + i yddot. The equation of motion becomes
This is a second order, linear, homogeneous differential equation. To find the solutions, try the solution η = e-i αt where α is a yet-to-be-determined constant. Substitute and cancel out the common exponential factor to find
which is satisfied for
The approximation assumes that the angular velocity of the pendulum is much greater than the earth's angular velocity, ω >> Ω. We have the required pair of independent solutions, so the general solution cna be written:
If the pendulum is started from rest with x=A and y=0, then the solution becomes, as you should find in your homework problem:
This solution says that the amplitude doesn't change. But since Ωz << ω0, we can break the motion down into a fast part and a slow part. The fast motion is the swinging of the pendulum, goverened by the cos(ω0 t) term. During one period of the pendulum, the slow part of the motion, the e-iΩz t term, barely changes. To see an appreciable change in the exponential piece, we must wait many minutes, or hours, while the period of the pendulum is typically a few seconds. So we can view the motion as the pendulum swinging in a plane, with a slow rotation of the plane of swing. The plane of swing is determined by the complex exponential, e-i Ωz t = cos(Ωz t) - i sin(Ωz t). The angle of the plane to the x-axis is Ωz t = ( Ω cosθ ) t, and the change of the angle is called the precession.
Let's verify that the precession is slow, everywhere on Earth. At the North or South Pole, cosθ = ±1, so to change the angle by 2π takes 24 hours -- recall that Ω ≈ 2π / (24 hr). At the equator, cosθ = 0; the pendulum doesn't precess! (It has an infinite period of precession.) At a location inbetween, like Detroit where θ ≈ 47°, the pendulum precesses, but it takes more than a day to precess 360°. In particular, it takes time T = 24hrs/cosθ which is about 35 hours at the latitude of Detroit.
Please read this section on your own. Taylor compares the centrifugal and Coriolis forces to the terms that arise when working with spherical coordinates.