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HUMS 4100 Oscillations

A drumhead (Talman, MCSD Denver)

Simple Harmonic Motion

There are many forms of oscillatory motion is nature: e.g.

Mass on Spring
Pendulum
Air track pendulum
Stringless pendulum
Ball in bowl
Fortunately, all can be simplified to very similar mathematics
First some (universal) nomenclature
Amplitude (A): size of oscillation
Cycle: a complete oscillation (back and forward)
Period (T): time for one cycle
Frequency f: no. of cycles/s = 1/T

The Spring

Easiest mathematically is the spring. One more special force:

CEIIINSSSOTTUV
which you will immediately realise is
SIC VIS UT TENSIO
which in turn you will translate as ...
As the Force, thus the extension or as we say
Hooke's Law
extension ∝Force,
or
F = -kx
Note Only the extension matters
Force is in opposite direction to extension

Force from Hooke's Law
```
 F = -kx 
```
x is extension (not the original length)
- sign to get the direction right
k is the spring constant
Watch the simulation
Then Newton's 2nd law can be used
F = ma = - kx
Warning
This is one place we ought to use calculus: however we can cheat! There is a close mathematical relation between rotational and oscillatory motion:
- S.H.M. is "half" of rotational motion.
- so we need to find a better way to describe circular motion
Basic Trigonometry
We need a little trig
- θ = a R
  For these relations, we can work in either of
  
  Degrees: Full circle = 360°
  Radians: Full circle = 2π
  
  2π radians = 360⁰
- A useful mnemomic
  SohCahToa
  
  Sin = opposite = d hypotenuse R Cos = adjacent = R-δ hypotenuse R Tan = opposite = d adjacent R-δ
Special Values

sin(45⁰) = sin(π/4) = 1/√2

cos(45⁰) = cos(π/4) = 1/√2

sin(30⁰) = sin(π/6) = 1/2

cos(60⁰) = cos(π/3) = 1/2

sin(90⁰) = sin(π/2) = 1

cos(90⁰) = cos(π/2) = 0

We often have a convenient simplification when the angles are small
θ = a/R ~ sin(θ) = d/R cos(θ) ~ 1
Note that this only works if we measure angles in radians:
Will frequently use "sine-wave" as convenient way to represent oscillations or waves in general: Note
Frequency f: no. of cycles/s
Angular Frequency ω no. of cycles/s
```
ω  = 2πf
```

Back to circular motion

Formally, we can write

\color{red}{ \begin{array}{l} x = A\cos \left( {\omega t} \right) \\ y = A\sin \left( {\omega t} \right) \\ \end{array}}

for circular motion (because θ = ωt).

Just use the y-part of the motion (it doesn't matter which we choose). get agreement if

\color{red}{ \omega ^2 = \frac{k}{m}}

or period

\color{red}{ P = 2\pi \sqrt {\frac{m}{k}} }

THis behaviour is universal: whenever we get a system with

acceleration = -constant * displacement

we can write it as
\color{red}{ a = - \omega ^2 x}
the solution is
\color{red}{ x = A\sin \left( {\omega t} \right)}
and Period
\color{red}{ P = \frac{{2\pi }}{\omega }}

Physical Pendulum

Look at the Foucault pendulum in the entrance to Herzberg building:

Watch the animation. In this case, tension in string supplies force to return bob to centre
a = -g x L

with the same solution as before
x = A sin(ωt)
or (better)
θ = θ₀ cos(ωt)

for a simple pendulum

```
ω² = g/L
```
Period: ω radians/s means
```
f = ω/2π cycles/s 
= ω/2π Hz
```

so the period is

P = 1  =  2π  =  2π(L/g)^1/2
    f      ω

What should period be for the Foucault pendulum? (L ~ 20 m.)
CHECK IT
What would period of a 1 m simple pendulum be?
Why do you walk with your arms extended, but jog with them bent?
What should the natural walking pace for an adult be (in steps/s)

P.E. for spring:

Force varies ⇒ P.E.

U = 1/2 k x²

Note that the P.E. is positive for x < 0 as well as x > 0: you can store energy in a stretched spring or a compressed spring.

Energy and S.H.M.

As usual, Tot. Energy = P.E. + K.E. is conserved.

P.E. = ¹/₂ k x² 
K.E. = ¹/₂ m v²
Total Energy = ¹/₂ k x² + ¹/₂ m v²

Damped S.H.M.

Reality rearing its ugly head again. If there is friction in the system, oscillations will die away. Formally, if the damping force F ∝ v, then the solution would look like

y = y₀e^-atsin(ωt)

The frequency is almost the same, but the amplitude decreases exponentially with time

The extreme cases are

a pendulum swinging in vacuum: almost undamped (a = 0.1 small)
a pendulum swinging in syrup: will not complete one swing (a = 1.0 large)

Forced oscillations and resonance

Up to now, we have assumed that the system is started off at some time and allowed to oscillate. However, can arrange for the system to be driven, so that driving force varies in time. Watch the demo:

If driving frequency ω₀ matches natural frequency of system ω, then size of oscillations grows

Usually some damping, so does not grow without limit. e.g.

Soldiers marching across bridge (did it ever happen?)
Status: RO
From: "MIDN Zamberlan" <m017107@usna.edu> To: <watson@physics.carleton.ca>
Subject: Answering a rhetorical question
Date: Fri, 17 Nov 2000 01:28:03 -0500 X-Priority: 3
I was reading your class on simple harmonic motion and you said "soldiers marching across bridge ,did it ever happen?" Well, actually I have seen it occur at the US Naval Academy. In '97 the freshmen, plebes, were out exercising and ran across a wooden footbridge in cadence. There was 50 of them. The bridge is about 200 ft long and once they reached the middle, two of the supports snapped and the bridge began to oscillate noticeably. The bridge was then condemned till a Naval Construction Battalion was able to repair it. I hope this answers your question.
Very respectfully,
RJ Zamberlan MIDN, USN

Bay of Fundy,
hot air over beer bottles,
musical instruments (most)
wash-board roads,
resonances in nuclei,
electrons in an AC circuit and...............................
the Tacoma Narrows bridge
Now we go to waves and sound