4.7 Simple Harmonic Motion

Many physical periodic happenings can be represented as a sinusoidal function * t is time * is amplitude * is period * is frequency (maximum displacement) (time for 1 cycle) (# of cycles per unit of time) Which do we choose? sine or cosine? - Whichever seems the most appropriate!

Some Examples (but not all the various kinds!) Ex 1) An alternating current can be described by a sinusoidal function where c(t) is the current measured in amperes at t seconds. Find the amplitude, period, & frequency. a bc amp = Per = frequency =60 18

Ex 2) The amplitude of a sound wave produced by the note E above middle C is 0.8 and the frequency is 330 cycles per second. Determine the sinusoidal function representing this sound. a = 0.8  Per = frequency = 330

Ex 3) A weight hanging from the end of a spring is pulled down 12 cm below its resting place & released. It takes 2.4 seconds to complete one cycle. Determine an equation of motion for the weight. a = 12 b = ? (12 down from rest) Use the cosine function Why cosine? Starts low… not in middle But it’s LOW instead of high How to fix this? Flip over x-axis Make equation negative 12 cm

Ex 4) A Ferris wheel is 80 ft. in diameter & rises 86 ft from the ground. Each revolution of the wheel takes 28 seconds. Express the height of a rider as a function of time t if the rider is at the bottom when t = 0. bottom … bottom not at 0  at 6 d = ft diam = 80 ft  radius = 40 ft 86 ft 6 ft 6 86 neg. cos = 46 (middle)

Homework #408 Pg 233 #1, 7, 9, 13, 15, 17, 19, 27, 30, 33, 37, 39