Linear Algebra, 2020a

486 Chapter Five. Similarity
That equation is quadratic in ω 2 . Apply the formula to solve quadratic equations,
(−b ± √ b 2 − 4ac)/(2a).
√
ω 2 = 2ω2 0 ± 4ω 4 0 − 4(ω4 0 − ɛ2 /4mI)
= ω 2 0 ± ɛ
2
2 √ mI
The value ɛ/ √ mI = ɛ/ √ κk is often written ω B so that ω 2 = ω 2 0 ± ω B/2. This
is the beat frequency, the difference between the two normal mode frequencies.
Although the argument is beyond our scope, the general formula for the
motion of the pendulum is a linear combination of the motions during the normal
modes. Thus, the pendulum’s motion is entirely determined by the eigenvalues
of the above matrix. See [Berg & Marshall].
Exercises
1 Use the formula for the cosine of a sum to give an even more general formula for
simple harmonic motion.
2 Find the eigenvectors associated with the eigenvalues.
3 Find the values of ω in the case where x(t) =A 1 cos(ωt+φ) and θ(t) =A 2 cos(ωt+
(φ + π)).
4 Build a Wilberforce pendulum out of a Slinky Jr and a soup can. You can drill
holes in the can for bolts, either two or four of them, that you can use to adjust
the moment of inertia of the can so the periods of vertical and rotational motion
coincide.