Understanding Oscillations in Physics
Explore the world of oscillations through concepts like Simple Harmonic Oscillators (SHO), damped oscillations, underdamped and overdamped systems, critically damped oscillations, and more. Learn about the properties of waves, the wave equation, and electrical oscillations in RLC circuits.
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Presentation Transcript
Simple Harmonic Oscillator (SHO) Newton s Second Law: ?? = ? ? ? ? Equations of Motion: ? + ?02? = 0, where ?0= Solution: ? ? = ?cos(?0? + ?) Useful Equation: ? = 2?? =2? ?
Damped Oscillations Newton s Second Law: ?? ?? = ? ? Equations of Motion: ? ?and ? = ? ? + 2? ? + ?02? = 0, where ?0= 2?
Underdamped: < 0 Solution: ? ? = ?? ??cos(?1? + ?) ?02 ?2 Where ?1=
Overdamped: > 0 Solution: ? ? = ? ??(???2?+ ?? ?2?) 2 ?2 ?0 Where ?2= Will not oscillate
Critically Damped: = 0 Solution: ? ? = ? ??(? + ??) Will reach equilibrium in the minimum amount of time.
Damped and Driven Oscillations Newton s Second Law: ?? ? ? + ?0cos?? = ? ? Equations of Motion: ? + 2? ? + ?02? = ? cos??, where ? =?0 ? Solutions: ? ? = ? ? + ??? ? ??? = ??cos(?? + ??), where ??= (?02 ?2)2+4?2?2
Resonance ?02 2?2 ??= Quality Factor ? ?? 2?
Electrical Oscillations RLC Circuit: Resistor: ??= ?? Capacitor: ??= ?/? Inductor: ??= ??? ?? EOM: ? ? + ? ? +1 ?? = ?0cos?? ? ? + ? ? + ?? = ?0cos??
The Wave Equation ??? ???= ????? ??? Principle of Superposition: For any solution to the wave equation of the form f(x vt) and g(x vt), the function (f+g) also solves the wave equation Example: f(x,t) = 1 2(cos ? + ?? + cos ? ?? )
The Wave Equation ??? ???= ????? ??? Principle of Superposition: For any solution to the wave equation of the form f(x vt) and g(x vt), the function (f+g) also solves the wave equation Example: f(x,t) = 1 2(cos ? + ?? + cos ? ?? ) Also expressed as ? ?,? = cos ? cos(??), which is more easily recognized as a standing wave.
Basic Equations ? ?,? = ? ???(?? ?? + ?) ? = ? ? ?? ? =? ? =2? ? =2? ?= 2?? A: amplitude k: wavenumber : angular frequency : phase ? ? White Book Problem (pg 122): What is the absolute value of the relative phase between two waves described by sin(x-vt+ /6) and cos(x-vt)?
Basic Equations ? ?,? = ? ???(?? ? + ) ? = ? ? ?? ? =? ? =2? ? =2? ?= 2?? A: amplitude k: wavenumber : angular frequency : phase ? ? White Book Problem (pg 122): What is the absolute value of the relative phase between two waves described by sin(x-vt+ /6) and cos(x-vt)? Answer: Inherent phase shift between sin and cos is ? so /2- /6= /3 2. Cos is shifted left of sin,
Phase and Group Velocity ??=? ? ??=?? ?? ? ?=? (True for EM waves. Note: the wavelength is modified, the frequency stays the same) ? Red dotted line shows the envelope of wave packet that travels at the group velocity. Individual crests travel at the phase velocity, which can be a function of k if (k). Therefore different wavelengths can have different speeds (dispersion).
Poynting Vector Using Maxwell s equations, we can find the solutions: k is the direction of propagation and n is the direction of polarization (only for E). Poynting Vector describes the transport of energy I is the intensity, or the average power per unit area
Maluss Law ? ?,? = E0 ??? ? ? ?? + ? ? ? = ?0???2 ? =1 2?0 ? =1 2?0 ? = 0
Maluss Law ? ?,? = E0 ??? ? ? ?? + ? ? ? = ?0???2 White Book Problem (pg 121): Polarized light with polarization vector n=2x+3y is incident on a polarizer oriented at v=x+2y. What is the ratio of the intensity of transmitted light to the initial intensity
Maluss Law ? ?,? = E0 ??? ? ? ?? + ? ? ? = ?0???2 White Book Problem (pg 121): Polarized light with polarization vector n=2x+3y is incident on a polarizer oriented at v=x+2y. What is the ratio of the intensity of transmitted light to the initial intensity ?? ?0 ? ? ? ? 2 + 6 2=64 = 2= 65 13 5
Brewsters Angle ??= tan 1?2 ?1 n1 n2
Interference ? ?,? = ????(?? ??) ? ?,? = ???? ?? ?? + Phase Shift ( ) Type of Interference Examples Constructive -Odd multiple of - 2m m=0,1,2 - Double-slit interference - Single-slit diffraction Destructive -Even multiple of - (2m+1) m=0,1,2 - Thin films *Most interference problems on the GRE rely strictly on memory.*
Double-Slit Interference Monochromatic point source ( ) Maxima ????? = ?? Minima ????? = (? +1 2)? d L
Single-Slit Interference Monochromatic point source ( ) Assume L>>d Minima ????? = ?? Maxima a L no simple formula central max at center
Diffraction by a Circular Aperture GRE only tests limiting case of the first diffraction minima. D First Circular Diffraction Minima D???? = 1.22? (Left) The resultant diffraction pattern of a circular aperture. (Right) A circular aperture.
Bragg Diffraction X-rays incident on a crystal lattice. Like double-slit, interfere by pld. Maxima d???? = ??/2
Geometric Optics Assorted Topics
Reflection & Refraction = i r = sin sin n n 1 = 1 2 2 index of refraction n
Lens Equation = 1 1 1 object distance d o + = = image distance d d d f i = focal length f o i
Lensmakers Equation 1 f 1 R 1 R = ( ) 1 n 1 2 = focal length f = = + 1 R R = radius of curvature R = index of refraction n 2
Rayleigh Scattering for a 4 6 I I a 0 = intensity I = wavelength = particle size a
Doppler Effect + v v = r f f 0 v v s = observed = frequency f emitted frequency f 0 = wave = velocity v = s , + & r v v r s receiver v elocity v = s , & r v v r r s = source velocity v s
Standing Sound Waves = 4 L = 2 L
