Physics 130 Formula Sheet – Stefan Martynkiw Simple Harmonic Motion Dampening F d=−bv x −b t x = A⋅cos t x = Ae 2m cos ' t =− A sin t v =− x
2
a x =− A cos t 2
a x =− x
= 2 f
k g = m L 1 k f = = 2 2 m 1 m T = =2 f k − v 0x = arctan x0
=
A= x 0
2
v0
2
2
Energy in SHM 1 1 1 2 2 2 E= m v x k x = k A 2 2 2 k 2 2 v x =± A − x A m v max occurs at x =0
2
k b − 2 m 4m For underdamped situations, b 2 km km , use the above x formula. In Critically damped situations, w' = 0. Energy in Damped situations. dE =−b v x 2 dt
' =
Forced Oscillations F max A= 2 2 2 2 k − m d b d 2 When k −d =0 , A has a k / m . maximum of d= k The height is proportional to 1/b. Wave Speed v = =⋅ ⋅f Wave Number 2 k = ; v =/ 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 v wave on string =
F
Rate of Energy Transfer for a wave P x ,t = Fk A 2 sin2 kx − t 2 2 P x ,t = F A sin 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: =// 4 , 3 / 4 , 5 / 4 , ... x =
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 Δ P 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:
I =
Power from source 2
4 πr Inverse square law 2 I 1 r 2 = 2 I 2 r 1 Intensity = Pressure X Velocity (relating intensity to either the displacement or pressure amplitudes).
Standing Waves in a Pipe Two open e nds 2L nv λ n= f n= n 2L One closed end (“Stopped”) n = 1,3,5, ... nv 4L λ n= f n= 4L n Phase Difference and Path difference.
Light Rays / Polarization I = I 0 cos θ Snell's Law sin θ 1 n 2 v 1
λ = = = λ1 sin θ 2 n 1 v 2 2
Phase difference is based on the creation of the wave at its source. Path difference is t he different distances the two waves must travel.
Relating the two:(assuming created in phase) Δ L Δ ϕ= ⋅2 π
λ
Beats T beat =
T a T b
T b −T a Doppler Effect f L =
Instantaneous Intensity 2 2 I ( x , t )= B ω k A sin ( kx −ω t ) Average Intensity , a is displ ampl 2 2 I =1 / 2 √ ρ B ω A
f beat =∣f a −f b∣ v ± v L v± vs
f s
2
2 √ ρ B
tan θ B=
nb
na Geometric Optics /Spherical Mirrors f = R / 2 hi −d i 1 1 1 = + m= = f d i d o ho d o Refraction with Spherical Boundary nair n glass nair −n glass
+
=
Lateral Magnification is 1.
Decibel Scale
nair
I β=( 10 dB) log10 ( ) , I 0 −12
At the Brewster angle, all reflected light is polarized. Where nb is the “water” in the textbook diagram.
di r curvature nglass r curvature −nair d o m= f = n glass−nair nglass d i Refraction at a plane
p max
I 0=10
n=c/v Refraction index Total Internal Reflection nb sin θcritical = , na= water na Polarization by reflection
do
Average Intensity of a sound wave in a fluid
I =
2
W / m
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 Half the Intensity After that: path length differs. D is space between holes 2 Path Length Difference
vs v
=−
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
I =4I o cos
1 2
( ϕ)
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
sin θ1=1.22
λ
Diameter
Rayleigh's Criterion ( resolution of two objects. The angle seperating the two objects.)
θ R=1.22 λ
Interference Intensity for Two “Wide” Slits
do di Lens-maker's Equation 1 1 1 =( n −1) − f R1 R 2
(
Constructive interference at
D
n glass
n = index of refraction R's = radii of curvature
Δ L =d sin θ Phase Difference ϕ=( d sin θ)⋅(2 π)/λ
)
2
(
ϕ sin α α
I = I m cos ( ) 2
)
2