Harmonic Motion
Old version
Key variables:
a = a(t)= amplitude
k= initial amp
c= damping constant
omega = h. stretch (or comp.)
Basic Example
#1: k=20, c=0.5, p=4, y=a(t)cos(omega*t)
Answer
Word Problems
Guitar
#2: A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2cm from its resting position. After 3 seconds, the displacement of string measures 1 cm. Find the damping constant.
Ans: ln(2)/3
Animal Population
#3: A deer population oscillates 19 above and below the average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 deer each year. Find a function that models population, P, in terms of months since January (t).
Ans: P(t)= -19cos([pi/6]t)+800 + 40t/3
Spring
#4 A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds and its amplitude decreases by 30% every second. Find an equation that models the position of the spring “t” seconds after geing released.
Ans: y= [-5e^(-0.357t)]cos([2pi/3]t)+10
#5
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Shortcut
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