BIT1002 Elasticity and SHM

By the end of this you should understand oscillating systems, which means understanding Hooke's Law Elasticity and Young's Modulus Simple Harmonic Motion The Simple Pendulum Damped and Forced Oscillations

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

P.E. for spring:

W.D. = F x

• δW = F dx which is the area of the green rect.

• So the total W.D. is given by area under line. Hence the P.E. is ..... 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.

A toy gun consists of a spring which fires a 40 gm. cork. The spring can be compressed 1.5 cm by a 0.5 kg mass. To fire the cork, the spring is pulled back 10 cm. What is k? What is the velocity when it is released How high will it go? If the spring is pulled back twice as far will its velocity Be twice as great Be four times as great Be 21/2 times as great? Be the same?

Elasticity

However, if a force is applied to a material, it will bend or break Simplest is Young's modulus: forces applied to each end of a bar will compress or stretch it

• Stress = Force/unit area	= F/A
  Strain = extension/original extension	= δL L0

• Independent of whether we have an extension or compression, (Stress) = E (Strain) Hooke's law

  E = F L0 A δL

Young's modulus

As usual, reality is more complex

Stress-strain plot will remain linear up to some max (A), then be distorted (B), then represent some region where the material becomes permanently distorted (C) up to some "ultimate strength"(D) where it will break

Compression and tension.

Young's moduli need not be the same

• Units: Stress is a form of pressure: Force/unit area = N m^-2 = Pascals
• (Pressure in meteorology: 1 atmosphere ~ 10⁵ P = 100 kP) Strain is dimensionless, so

• [E] = N m-2
  

• e.g how much would a 1.5 m long piano wire 0.3 mm in diameter stretch if you suspend a 5 kg weight from it? (E = 2x10¹¹)

  δL = F L0
       E A 
  

Ultimate strength

Breaking strain depends very much on whether material is stretched, compressed, sheared or twisted. Units also N m^-2

• Material	Tensile	Compressive	Shear
  Steel	5x108	5x108	2.5x108
  Concrete	2x106	2x107	2x106
  Wood	4x107	3.5x107	5x106
  Bone	1.3x108	1.7x108

• e.g. bone has E = 1.6x10¹⁰, how much could Procrustes have stretched one's bones before they broke?

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

• Amplitude (A): size of swing
• Cycle: a complete oscillation (back and forward)
• Period (T): time for one cycle
• Note that there are two commonly used units for rotational/vibrational motion
• Frequency f = number of turns (vibrations)/s = 1/T usually called Hertz (Hz)
  
• Angular velocity ω (= no of radians/sec) Relation is obvious: \color{red}{\omega = 2\pi f}
• Hertz also used for wave motion

Springs

• 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

• Then Newton's 2nd law can be used

  F = ma = - kx
  

Warning

• There is a close mathematical relation between rotational and oscillatory motion: S.H.M. is "half" of rotational motion.

• S.H.M. is "half" of rotational 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 x-part of the motion (it doesn't matter which we choose).
• Then the velocity is the deriv:
  \color{red}{ v_x = \frac{d}{{dt}}A\cos \left( {\omega t} \right) = - A\omega \sin \left( {\omega t} \right)}

• and the acceleration is
  \color{red}{ a_x = \frac{{dv_x }}{{dt}} = - \frac{d}{{dt}}A\omega \sin \left( {\omega t} \right)}
• or
  \color{red}{ a = - A\omega ^2 \cos \left( {\omega t} \right)}
• but we want this to satisfy Newton's 2nd law
  \color{red}{ ma = - kx}
• so
  \color{red}{ m\left( { - A\omega ^2 \cos \left( {\omega t} \right)} \right) = - k\left( {A\cos \left( {\omega t} \right)} \right)}
• which is O.K. provided
• \color{red}{ \omega ^2 = \frac{k}{m}}

1. 20 rads/s
2. 400 rads/s
3. .0025 rads/s

1. 3.18 s
2. 126 s
3. .314 s
4. .008 s

1. 1
2. 1/2
3. 1/4
4. 2
5. 4

F = -kx

Energy and S.H.M.

• What is P.E. for spring? In oscillation P.E. = 1/2 k x2 K.E. = 1/2 m v2 Total Energy = 1/2 k x2 + 1/2 m v2

• At max extension:

  P.E. = 1/2 k A2 
  K.E. = 0
  

• At equilibrium point F = 0, so

  P.E. = 0 
  K.E. = 1/2 m v02
  

• Hence

  v02 = (k/m)A2
  

• and can get speed at any intermediate point:

Physical Pendulum

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)

• 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?

1. The natural rythmn of walking corresponds to your arms acting as pendulum.
2. It helps to have your hands below waist-level
3. It is a evolutionary leftover from when we walked with our hands touching the ground.
4. It is psychologically more natural

1. .1
2. .5
3. 1
4. 2
5. 5

Would a pendulum clock run fast or slow on the moon?

1. fast because gravity is stronger
2. fast because gravity is weaker
3. slow because gravity is stronger
4. slow because gravity is weaker
5. The same time because time on the moon is the same as here

Damped S.H.M.

Formally, if the damping force

F ∝ v, then the solution would look like

y = y0e-atsin(ωt)

• 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

• k = 400 m = 10 so ω₀ =(k/m)^1/2 = .....?

• If driving frequency ω matches natural frequency of system ω₀, then size of oscillations grows
• Usually some damping, so does not grow without limit

Forced oscillations and resonance

• 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,
• wash-board roads,
• resonances in nuclei,
• electrons in an AC circuit and...............................

• the Tacoma Narrows bridge

A =    A0    
   (w-w0)2+ γ2

• What does this look like? For γ= .1, .2, .5

• It is easier to see this if we make them all have the same height

• Now we go to waves and sound