Physics 130 Formula Sheet 3x766

Physics 130 Formula Sheet – Stefan Martynkiw Simple Harmonic Motion Dampening F d=−bv x −b x= A⋅cos t t 2m x= Ae cos ' t v x =− A sin  t k b2 a x =−2 A cos  t '= − 2 2 m 4m a x =− x For underdamped situations, b2  km , use the above x =2 f formula. In Critically damped k g = = situations, w' = 0. m L Energy in Damped situations.  1 k dE 2 f= = =−b v x 2 2 m dt 1 m T = =2  Forced Oscillations f k F max −v 0x A= =arctan  x0  k−m d2 2b2 2d When k −2d =0 , A has a v 02 2 A= x 0  2 maximum of  d=  k /m .  The height is proportional to 1/b. Energy in SHM Wave Speed 1 2 1 2 1 2 E= mv x  k x = k A v =⋅f 2 2 2 Wave Number k 2 v x =± A 2−x 2  k= ; v =/k m  v max occurs at x=0

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Mechanical Waves Wave function to the right y  x ,t =Acos kx− to  Wave function to the left y  x ,t =Acos kx to 

Metric Prefixes

Linear Mass Density: mstring = L F v wave on string = 

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Rate of Energy Transfer for a wave P x ,t =Fk  A 2 sin 2  kx− t P x ,t =  F  A 2 sin 2 kx− t  Standing Wave Frequencies Pavg=1 /2 Pmax This v is speed of wave on a string. v f n= Standing Waves n y standing  x , t =Asin kx sin  t Fundamental frequency for a string Shape at a position depends on fixed at both ends: sin(kx); Shape at a time depends on 1 F f 1= sin(wt) 2L  Nodes: x=0 , /2 , , 3 /2 ,... Antinodes: x=/4 , 3 /4 , 5 /4 , ...

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Allowed wavelengths for a standing wave on a string with nodes at x=0, x=L 2L  n= n

Sound Waves Pressure Formulas Bulk Modulus ΔP B= Δ V /V Difference in atmospheric pressures in a sinusoidal soundwave:

p(x ,t )=BkAsin (kx −ω t) pmax =BkA=(v ρω) A Speed of Sound in a fluid: B v = ρ , rho is the mass density Intensity I =Pressure/ Area

√

Intensity of sound in spherical waves:

Power from source I= 4 π r2 Inverse square law I 1 r 22 = I 2 r 21

Intensity = Pressure X Velocity (relating intensity to either the displacement or pressure amplitudes).

Instantaneous Intensity 2 2 I (x , t)=B ω k A sin (kx −ω t) Average Intensity, a is displ ampl I=1/2 √ ρ B ω2 A 2 Average Intensity of a sound wave in a fluid

p2max I= 2 √ρ B Decibel Scale β=(10 dB) log 10 ( I 0=10−12 W /m2

Standing Waves in a Pipe Two open ends 2L nv λ n= f n= n 2L One closed end (“Stopped”) n = 1,3,5, ... 4L nv λ n= f n= n 4L Phase Difference and Path difference. Phase difference is based on the creation of the wave at its source. Path difference is the different distances the two waves must travel.

Light Rays / Polarization 0

Snell's Law sinθ 1 n 2 v 1 λ 1 = = = sinθ 2 n 1 v 2 λ 2 Refraction index n=c/v Total Internal Reflection n sin θcritical = b , na=water na Polarization by reflection

Relating the two:(assuming created At the Brewster angle, all reflected light is polarized. Where nb is the “water” in the in phase) textbook diagram. ΔL n Δ ϕ= ⋅2 π tan θ B= b λ na Beats Geometric Optics /Spherical T T f beat =∣f a −f b∣ T beat= a b Mirrors T b −T a f =R/2 Doppler Effect h −d 1 1 1 = + v±v L m= i = i f di do f L= f ho d o v±v s s Refraction with Spherical Boundary nair n glass nair −n glass + = do di r curvature nglass r curvature −n d f= m= air o n glass−nair nglass d i Refraction at a plane Lateral Magnification is 1.

I ) , I0 Sonic Booms and Shockwaves Shockwave Angle:

v sin θ= vs

vs Mach Number = v

Interference

Unpolarized light entering the first polarizer -> In Young's double-slit experiment, only the Half the Intensity After that: path length differs. D is space between holes 2 Path Length Difference I =I cos θ

nair n =− glass do di Lens-maker's Equation 1 1 1 =(n−1) − f R1 R 2

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n = index of refraction R's = radii of curvature

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Δ L=d sinθ Phase Difference ϕ=(d sinθ )⋅(2 π)/λ Constructive interference at

ϕ=2 π m ,(m=0,±1,±2, ...) Destructive interference at

ϕ=2 π( m+1/2),(m=0,±1,±2,...) Fringe locations can be found by combining the above 3 formulas (whether for constructive or destructive) Two Source Intensity Io = intensity of each source 2 1 I =4I o cos ( ϕ) 2

Diffraction Any pair of rays seperated by a/2 has the same phase difference. “a” is width of hole Dark fringes at

a sin θ=m λ , m=±1,±2

Single Slit diffraction intensity

sin α 2 , Im is max intensity I (θ)=I m α α=1/2 ϕ= π α sin θ λ

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Circular Aperatures Location of first dark fringe

sin θ1=1.22

λ Diameter

Rayleigh's Criterion (resolution of two objects. The angle seperating the two objects.)

θ R=1.22 λ D

Interference Intensity for Two “Wide” Slits

ϕ sin α I =I m cos 2 ( ) α 2

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