Harmonic Motion Lesson 2.8

2  Consider a weight on a spring that is bouncing up and down  It moves alternately above and below an equilibrium point  The movement can be modeled by The Spring Has Sprung

3 Simple Harmonic Motion  For the functions t is time f is the frequency 1/f is the period |a| is the amplitude

4 Try It Out  For each of the following, find the Amplitude Frequency Period

5 Try It the Other Way  Given Frequency =.8 cps Amplitude = 4  Write the function  What if Amplitude = 3.5 Period = 0.5 sec Assume maximum displacement occurs when t = 0

6 Spring Constants  For a particular spring system When mass = m When spring constant = k  The frequency is calculated Given k and m, substitute into function See exercise 28, page 172

7 Damped Harmonic Motion  What if the a is not a constant Rather it is a function As time, t increases, the motion is lessened by a dampening influence  Experiment with spreadsheetspreadsheet  Where is dampening important on an automobile?

8 Damp Your Motion  Given  How many complete oscillations during time interval 0 ≤ t ≤10  How long until the absolute value of the displacement is always less than 0.01  Hint: use calculator

9 Damp Your Motion  Count oscillations for 0 ≤ t ≤10  For when movement is less than.01 zoom in draw lines at y = ±.01

10 Damp Your Motion  Double check values at the peak  Ask calculator for intersections

11 Assignment  Lesson 2.8  Page 208  Exercises 1 – 35 odd