Simple Harmonic Oscillator - Motion

• Equation of motion for SHO.• Motion animation.• Sinusoidal solution and harmonic frequency.• Terminology and summary.• Resonant frequency animation.• Example problems.• Relation between vmax , a ax , and A.• Problem strategy.• Simple pendulum.

Equation of Motion

– Given the following:

– What is equation of motion?• • Of course not!• Must be something oscillatory!!

Compare with Circular Motion

Compare Simple Harmonic and Circular Motionhttp://www.animations.physics.unsw.edu.au//jw/SHM.htm

Simple Harmonic vs. Circular Motion

• Simple Harmonic vs. Circular Motion Animation• http://www.animations.physics.unsw.edu.au//jw/SHM.htm

• See Appendix for running embedded files.

1. To view web link put Powerpoint in reading view.2. To view embedded file put Powerpoint in normal view.

shm_spring1.swf

Sinusoidal Solution 1

• Must solve equation:

• Try this form:

• Reason:– Goes back and forth like animation– its own 2nd derivative– extra terms give flexibility

• Derivatives– If – then – and

Sinusoidal Solution 2

• Must solve equation:

• Plug in on both sides

• Wonderful things happen– The “sin” terms cancel.– The minus signs cancel

• Result– Both sides track each other (sine or cosine)– Natural resonant frequency

Resonant frequency• Resonant frequency

• Animationhttp://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html

mass-spring-lab.swf

Harmonic Oscillator Terminology

• Cycle – One complete oscillation

• Amplitude – Endpoint limits (x = -A to +A)

• Period – Time to complete one cycle

• Frequency – Number of cycles per second

• Frequency vs. Period– f = 1/T

– T = 1/f

SHO - summary to date• Energy

• Motion

• Harmonic frequency

• Frequency and period

Example – Problem 7

• m, k, ω – if you know 2/3 you can always find 3rd

• For m = 0.0005 kg

Example – Problem 9 (I)

• m, k, ω – if you know 2/3 you can always find 3rd

• Total Energy – Just find potential at full amplitude

• Velocity at equilibrium point

Example – Problem 9 (II)

• Velocity at 0.1 m

• Starting condition: at

Must use cosine!

Example – Problem 13 (I)

• At any point x

• Amplitude

• Max velocity

Example – Problem 13 (I)

• Resonant frequency

• Equation of motion?

Since it doesn’t start at either equilibrium or full amplitude, this requires phase angle

We don’t do in this course – to complicated!

Example – Problem 19 (I)

• Oscillation is given in terms of period

• Starting condition: at 0.18

Must use cosine!

• Will reach equilibrium after ¼ cycle

Example – Problem 19 (II)

• For maximum velocity

• For maximum velocity (at full amplitude)

Solving SHO problems• If stretched/compressed and release from rest, then you

know amplitude and total energy.

• If velocity known at equilibrium midpoint, then you know vmax and total energy.

• If you know total energy, you can subtract potential or kinetic to get the other.

• If you know k and m, you know ω.

• General form x = A sinωt or x = A cosωt

• If oscillation start at equilibrium sine, full-amplitude cosine.

vmax , amax , and A

• Vmax vs. A

(calculus)

• amax vs. A

(calculus)

Simple Pendulum• From Physics 103

• For ϴ small and in radians

• From geometry

• Resonant frequency is

• Acceleration is

• Problem 32 (f=0.572 Hz, E = mgl(1-cosθ)

Appendix - Animations• I use animations in this course as I think they are helpful. For each animation you can either click

on the web link or, if it’s unavailable, click on the embedded file directly.

• To use embedded files you have to have the Flash player standalone version loaded. To do this:– Download file flashplayer_11_sa.exe and put it in known location on your computer.– Hit Start menu, type “Control” in the search box, and this will open Control Panel.– Select “Programs”, “Default Programs”, and select “Associate a file type or protocol with a program”.– Wait forever for the darn thing to load.– In the list of file types, scroll down to “.swf”– Highlight “.swf” and hit “Change Program”, then “Browse”– Navigate to where you put flashplayer_11_sa.exe, and hit OK– Finished

• For non-ITS computers (store-bought configuration) just click on the file and it will prompt you.

• Mobile– The will run on Android, but not the Chrome browser.– I hear there’s a browser for iOS that will also work (Photon Flash Player)