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 v max , a ax , and A. • Problem strategy. • Simple pendulum.
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Simple Harmonic Oscillator - Motion Equation of motion for SHO. Motion animation. Sinusoidal solution and harmonic frequency. Terminology and summary.
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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
• 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)