More Oscillations! (Today: Harmonic Oscillators)
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1 More Oscillations! (oday: Haronic Oscillators) Movie assignent reinder! Final due HURSDAY April 20 Subit through ecapus Different rubric; reeber to chec it even if you got 00% on your draft: Springs as haronic oscillators. Fs - x Spring wants to push bac to equilibriu position. Graphing the Motion of Springs he paper oves at a constant speed underneath the pencil. If we were to graph what we observe, what would the position versus tie graph loo lie? A. B. C. D. Q3
2 oday s Main Ideas Siple haronic oscillators. Position, velocity, and acceleration with tie. Relation to pendulus. Siple Haronic Motion Any vibrating syste with F proportional to -x lie Hooe s law (F-x) undergoes SHM Called a siple haronic oscillator (SHO) Exaples: Spring; pendulu (for sall aplitudes), a person on a swing, vibrating strings, sound (next few lectures), a car stuc in a ditch being ``roced out! Pendula vs. Springs
3 Springs as haronic oscillators. ie it taes for one cycle ( period ): Note: not dependent on aplitude A! Period and Frequency of a Spring Period Frequency ƒ he frequency, ƒ, is the nuber of coplete cycles or vibrations per second Units of frequency! ƒ Units: /s or Hz (Hertz!) Side View of Circular Motion ω Bug ƒ Motion around a circle as viewed fro the side has a the sae position dependence as a spring
4 ω Bug Q4 ƒ For a spin rate of f 00 Hz, (00/s or 00rev/s) what is the bug s angular velocity, ω, in radians per second? A. 00 B. 00/() C. (00) D. 2(00) Angular Frequency Explicit definitions for SHM: he frequency gives the nuber of cycles per second ƒ he angular velocity/speed (or angular frequency) gives the nuber of radians per second ω 2 πƒ Motion as a Function of ie Use of a reference circle allows a description of SHM over tie! x is the position at tie t x varies between +A and -A Graphical Representation of Motion x When x is a axiu or iniu, velocity is zero When x is zero, the speed is a axiu (slope of x) v Acceleration vs. tie is the slope the of velocity graph. When x is ax in the positive direction, a is ax in the negative direction a
5 Suary of Forulas ω f x Acosωt v Aω sinωt 2 a Aω cosωt A a ax Calculator Warning! x Acosωt v Aω sinωt 2 a Aω cosωt What are the units of ω t? hus, your calculator will either need to be in radians to give the correct answer, or you need to convert ω t to degrees. radians 360 Calculator Chec You ve connected a 0g ass to a horizontal spring on a frictionless surface, and pull it out to a distance 0. fro its equilibriu point. You then let go and see that it does one whole oscillation once every 5 seconds. What is its x position after 23s? ω f x Acosωt v Aω sinωt 2 a Aω cosωt Siple Pendulu Copared to a Spring
6 he Siple Pendulu Gravity causes restoring force for oscillations. If θ is sall (sall aplitude oscillations): What causes it to swing bac and forth? F pendulu g x Should loo failiar! Fs - x Pendulu Siple Haronic Motion F pendulu g x Restoring force is proportional to negative of displaceent (F spring -x) Effective spring constant is eff g/ spring eff pendulu g he period of siple pendulu is independent of ass or aplitude; instead depends ONY on the length of cord! A siple pendulu has ass 2 g and length. What is the period of the pendulu? A) 2.0 s B) 2.8 s C) 4.4 s D) 8.9 s E) 9.7 s spring eff pendulu g Q5 Note: Daped Oscillations Why does a child stop swinging if not continuously pushed? When wor is done by a dissipative force (friction or air resistance), not all of the echanical energy is conserved. his eans not all of her potential energy at the top of each swing is converted into inetic energy so her next swing is not as high. he period of oscillations stays the sae. he aplitude decreases with tie.
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