Exercises for Chaos Under Control

Chapter 4: Noise, Noise, Noise

1. (a) A grandfather clock can be modeled by a simple pendulum, with period given by

T =2*pi*sqrt(L/g)

How long should the arm of the grandfather clock be in order for its "ticks" to be exactly 1 second long. (If you've ever observed such a clock you know that a "tick" is the duration between extreme positions of the bob, that is, the length of a tick is 1/2 the period of the bob's motion.) Take g to be 9.8 meters/ second2.

(b) At sea level g = 9.80 meters/second2 and at the top of a high mountain g = 9.78 meters/second2. A grandfather clock that has a 1.000 second tick at sea level has what duration tick at the top of the mountain?

(c)[E] Experimentally test the given relation between T and L by making a pendulum of a given length and timing its period. Hang a small mass from a string and tie the string firmly to a support. Draw the mass aside a small distance and let go. The time T is the interval between successive maximum displacements on the same side of the swing. The length of the pendulum is the distance from where the string is tied to the support to the center of the hanging mass (if the mass is a nice regular object like a sphere or cylinder; if it's not make an estimate of where the center of the mass is). Repeat your measurements for several different lengths.

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