Simple Harmonic Motion (SHM) 01 - Introduction

OPENING QUESTIONS: We have a video OQ today. Take a look at this animation and

1) Use that animation to come up with a definition for SHM.

2) What is the relationship between the distance the spring compresses (x) or is extended (also x) and the AMPLITUDE (wave height) of the descriptive wave?

3) How NASTY would this calculation become if we had to account for things like friction? (Hence the term SIMPLE harmonic motion)

OBJECTIVE:  

1)

WORDS/FORMULAE FOR TODAY:

  • (0) F = -kx
  • (1) x(t) = Acos (ωt + φ)
  • (2) ω = angular frequency = (k/m)1/2 measured in radians/second
  • (3) T = period = 2π/ω = 1/f
  • (4) f = frequency 1/T = measured in cycles per seconds or waves/sec
  • (5) v = dx/dt = -ωAsin(ωt + φ) (notice the similarity to rω)
  • (6) a = dx/dt = -ω2Acos(ωt + φ)(notice the similarity to rω2)

WORK O' THE DAY: 

Let's start by a digestions of terms:

Let's take 5 minutes to memorize formulae 0 - 4 please

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Our discussions of SHM will primarily deal with one dimension (x) so we'll typically ASSUME that any motion is constrained to X.

We can accurately predict the motion of an object experiencing SHM by the following formula:

x(t) = Acos (ωt + φ)

That formula characterizes the mathematical (graphic) representation of an object experiencing SHM

  • x = x direction
  • (t) = with respect to time
  • A = the AMPLITUDE of the resulting wave when the motion of the object is being graphed (measured in meters)
  • ω = angular frequency (measured in radians/sec)
  • t = time (in seconds)
  • φ (spelled "phi" and pronounced "fee") = the "phase constant" or "initial phase angle" and is measured in radians

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The amplitude of a wave is simply the HEIGHT of the wave... easy enough. However the HEIGHT of the wave is actually the "x" distance the spring compresses (or extends) too!

Let's take another gander at an example of SHM--

The two wave forms are slightly different... how?

xc

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The two waves are "out of phase". Notice that waveform "A" reaches its highest amplitude JUST BEFORE t = 0.

Waveform B is the SPECIAL case where the wave reaches its MAXIMUM amplitude at t = 0.

We can ADJUST waveform A to match waveform B by introducing a phase constant φ

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Let's take a swing at digesting our basic SHM wave equation:

x(t) = Acos (ωt + φ)

if A is the Amplitude of the wave:

  • What would ωt represent?

  • What would φ represent?

  • What would x(t) represent?

Perhaps some graphing would be helpful.... whip out some graph paper and take a gander at how x(t) changes (try it first on your own)

 

My suggestions (wolframalpha.com may be helpful here):

  1. Try graphing for the case where t = 0 and there is no phase constant
  2. Try graphing for the case where t = 1, ω = π/2 rad/sec and there is no phase constant
  3. Try graphing for the case where t = 2, ω = π/2 rad/sec and there is no phase constant
  4. Try graphing for the case where t = 3, ω = π/2 rad/sec and there is no phase constant

What is that equation telling us?

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Take a gander at Example 15.1

    • review our formulas o' the day with your group first
    • Now DO it

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Review 15.2 (it's not QUITE as nasty as it might first appear)

Pay special attention to the relation ship between T and f

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HOMEWORK:

 Problems: (Chapter 15 probs begining on page 474)

#1, #2, 3, 4, 10

 
STUDY GUIDE: