Chapter13 - Chapter 13 Lecture Essential University Physics...

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Unformatted text preview: Chapter 13 Lecture Essential University Physics Richard Welfsen 2nd Edition Oscillatory Motion Oscillatory Motion - A system in oscillatory motion undergoes repeating, periodic motion about a point of stable equilibrium. Hulh Iliulium haw the sum: period T - Oscillatory motion is {whim—Hm I characterized by Trainee-2r r—J- — Its frequency for period T=1/f. Period *l — Its amplitude A, or maximum excurSIon from equmbrlum. in Position - Shown here are position- versus-time graphs for two different oscillatory motions lg'gmfl fuel- with the same period and W :fllffii:fifm amplitude. A ' ' Time Position :3 Simple Harmonic Motion - Simple harmonic motion (SHM) results when the force or torque that tends to restore equilibrium is directly proportional to the displacement from equilibrium. — The paradigm example is a mass m on a spring of spring constant k. - The angular frequency co = 2afor this system is J? m: E — In SHM, displacement is a sinusoidal function of time: I{F:]=AEDE{E1H+¢5:] Displacement. .r — Any amplitude A is possible. _,.. - In SHM, frequency doesn’t depend 'l'iilwniii . "'L‘l't'nLL'l]. lll'.:_ —.' . op “amplitude. “ "F-‘u’lr‘d’l 'innpnfinl: “a. : ru:_l_ an. n Quantities in Simple Harmonic Motion - Angular frequency, frequency, period: m=EEf=JHm Iii/E 22%:3‘]? - Phase — Describes the starting time of the displacement-versus- time curve in oscillatory motion: x(t) = Acos(ra t + gal) Displacement, J: Other Simple Harmonic Oscillators - The torsional oscillator — A fiber with torsional constant arc provides a restoring torque. — Frequency depends on K'Ell'ld rotational inertia: _ - Simple pendulum — Point mass on massless cord of length L: J? E T=2E — t 3 t I : mL - l"ir:|'.'its=.linlt;ll r'nniL' 7|l'l‘-H'.l|.1|._‘l:_'3~ i11l!l'-L]l|-L‘ I'Jntiuzliludu 3* U a T=£=EE i moiminii'. Ell mg}; ']'h:n:‘.~; no torque f'l'nm ll‘iL: lunh'un it Iwi'unw Ii :1th “innit: this. linL! in tin: pit-til. SHM and Circular Motion - Simple harmonic motion can be viewed as one component of uniform circular motion. — Angular frequency a) in SHM is the same as angular velocity (u in circular motion. [an I [hi I r I As the position vector F traces out a circle, its x— and y— components are a sinusoidal functions of time. Energy in Simple Harmonic Motion In the absence of nonconservative forces, the energy of a simple harmonic oscillator does not change. — But energy is transfered back and forth between kinetic and potential lincrg :r' forms. 1 E E=Tizd Clll.'l'}_'_1-' f' it run-Hill". .. _ . taili :' |1I||I.‘l"!'.'l| L'I'L'|'_-'._'~ |'-' i anJ kill:l£;£|i:.‘l'_ll_";.' H-flifllléfl. Energy 'I'ime Pentium l L" K I i h-fl'lj'tl-jI-Efl [I H # we 3 imam L" K . lfl'fliiilj . -\._I F I _ w: w lilifliilifl if K - liliiigflfl ."II J] . Ila-J ' W - li.i::i:.-i;i-:1t:J u liftiii-iita-ii‘d'lj i," a" I I F I l' -— [J m = Muser-1E L" K Equihbrium .I. = U TEEIIIIS'F'un-u-Edmultn I1.- Damped Harmonic Motion - With nonconseryatiye forces present, SHM gradually clamps out: m diff : all: air {it — Amplitude declines exponentially toward zero: — For weak damping b, oscillations still occur at approximately the undamped frequency — With stronger damping, oscillations cease. - Critical damping brings the system to equilibrium most quickly. Thu object still oscillates sinusoidally . . . L... E s s U _ E % Tli‘i‘ic, i" E U E El E all a A ' ' _ _ _ . . . but llic illll|3]llL|Ll|: Llcci'caiscs (a) Underdamped, Grltlcally dampEd, williiii [l1u“u:n-u|npu"ul'u and (c) oyerdamped oscillations. “mil”?L‘llllillmlml Resonance - When a system is driven by an external force at near its natural frequency, it responds with large-amplitude oscmaflons. — This is the phenomenon of resonance. — The size of the resonant response increases as damping decreases. — The width of the resonance curve (amplitude versus driving frequency) also narrows with lower damping. till” (1' J: m t =—t::— b—+Fficosmdr air air Resonance curves for several damping strengths; we] is the undamped natural frequency filk/m Amplitude. rt ['J'J'H 2m“ 3-H)” Driving I‘rcqucncy. rod ...
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This note was uploaded on 02/12/2012 for the course PHYSICS 104 taught by Professor Staff during the Fall '10 term at Rutgers.

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Chapter13 - Chapter 13 Lecture Essential University Physics...

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