Physics (Phys 130): Formula Summaries And Review Sheet PDF

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Physics 130 Formula Sheet – Stefan Martynkiw Dampening Simple Harmonic Motion F d=−bv x −b x= A⋅cos t  t x= Ae2m cos ' t   v x =− A sin  t  2

a x =− A cos  t  2 a x =− x =2 f k = = m 1  = f= 2 2 1 T = =2  f −v 0x =arctan  x0

 



 

A= x 02

v0

g L

 

k m m k

2

2



Energy in SHM 1 2 2 1 2 1 E= m v x  k x = k A 2 2 2 k v x =±  A 2−x 2 m v max occurs at x=0





2

b k − m 4m 2 For underdamped situations, b2 km , use the above x formula. In Critically damped situations, w' = 0. Energy in Damped situations. dE 2 =−b v x dt  '=

Forced Oscillations F max A=  k−md 2 2b2 2d 2 When k −d=0 , A has a maximum of  d= k /m . The height is proportional to 1/b. Wave Speed v =⋅f Wave Number 2 ; v=/k k= 

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: m string = L F v wave on string = 



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  Pavg=1 /2 Pmax Standing Waves

Standing Wave Frequencies This v is speed of wave on a string. v f n= 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 2 4 πr Inverse square law 2 I 1 r2 = 2 I 2 r1 I=

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

p 2max 2√ ρ B Decibel Scale I=

I 0=10

W /m

Light Rays / Polarization

0

Phase difference is based on the creation of the wave at its source. Path difference is the different distances the two waves must travel.

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 nb Δ ϕ= ⋅2 π tan θ B= λ n a Beats Geometric Optics /Spherical Ta T b f beat =∣f a −f b∣ T beat = Mirrors T b −T a f =R/2 Doppler Effect 1 1 1 hi −d i = + v± vL m= = d d f f L= f ho d o i o v± vs s Refraction with Spherical Boundary nair n glass nair −n glass = + di r curvature do −nair d o nglass r curvature m= f= nglass d i n glass −nair Refraction at a plane

I ) , I0

2

Sonic Booms and Shockwaves Shockwave Angle:

sin θ=

v vs

Mach Number =

Interference

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

Lateral Magnification is 1.

β=(10 dB)log 10 ( −12

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.

vs v

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

(

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

)

Δ L=d sinθ Phase Difference ϕ=(d sin θ )⋅( 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

(

)

2

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

Circular Aperatures Location of first dark fringe

λ sin θ1=1.22 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

sin α 2 ϕ I = I m cos ( ) α 2

(

)

2...