Physics 130 Formula Sheet – Stefan Martynkiw Simple Harmonic Motion

x=A⋅cos tv x=− A sin t

ax=−2 A cos t

ax=−2

x

=2 f

= km

= gL

f =

2=

12 k

m

T=1f=2 m

k

=arctan −v0x

x0

A=x02

v02

2

Energy in SHM

E=12

mv x2

12

k x2=

12

k A2

vx=± km

A2−x2

vmax occurs at x=0

DampeningFd=−bv x

x=Ae−b2m

tcos ' t

'= km

−b2

4m2

For underdamped situations, b2km , use the above x

formula. In Critically damped situations, w' = 0.Energy in Damped situations.

dEdt

=−b v x2

Forced Oscillations

A=Fmax

k−md22b2

d2

When k−d2=0 , A has a

maximum of d=k /m . The height is proportional to 1/b.

Wave Speedv=⋅f

Wave Number

k=2

;v=/k

Mechanical WavesWave function to the right

y x ,t =Acos kx− to

Wave function to the lefty x ,t =Acos kx to

Linear Mass Density:mstring= L

v wave on string= F

Rate of Energy Transfer for a wavePx ,t =Fk A2sin2kx− t Px ,t = F A2sin 2

kx− t Pavg=1 /2 Pmax

Standing Waves

ystanding x , t =Asin kx sin t Shape at a position depends on sin(kx); Shape at a time depends on sin(wt)Nodes: x=0 ,/2 , ,3/2 ,...Antinodes:

x=/4 ,3 /4 ,5/4 , ...

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

n=2Ln

Metric Prefixes

Standing Wave FrequenciesThis v is speed of wave on a string.

f n=vn

Fundamental frequency for a string fixed at both ends:

f 1=1

2L F

Sound WavesPressure FormulasBulk Modulus

B=Δ P

ΔV /VDifference 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:

v=√ Bρ , rho is the mass density

IntensityI=Pressure/ Area

Intensity of sound in spherical waves:

I=Power from source

4 π r2

Inverse square lawI 1

I 2

=r2

2

r12

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

Instantaneous IntensityI (x , t)=Bωk A2 sin2

(kx−ω t)Average Intensity, a is displ ampl

I=1/2√ρBω2 A2

Average Intensity of a sound wave in a fluid

I=pmax

2

2√ρBDecibel Scale

β=(10dB) log10(II 0

) ,

I 0=10−12W /m2

Standing Waves in a PipeTwo open ends

λn=2Ln

f n=nv2L

One closed end (“Stopped”)n = 1,3,5, ...

λn=4Ln

f n=nv4L

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.

Relating the two:(assuming created in phase)

Δϕ=Δ Lλ

⋅2π

Beats

T beat=Ta T b

Tb−T a

f beat=∣f a−f b∣

Doppler Effect

f L=v±vL

v±vs

f s

Sonic Booms and ShockwavesShockwave Angle:

sin θ=vvs

Mach Number=vs

v

Light Rays / PolarizationUnpolarized light entering the first polarizer -> Half the Intensity After that:

I=I 0cos2θ

Snell's Lawsinθ1

sinθ2

=n2

n1

=v1

v2

=λ1

λ2

Refraction index n=c/vTotal Internal Reflection

sinθcritical=nb

na

, na=water

Polarization by reflectionAt the Brewster angle, all reflected light is polarized. Where nb is the “water” in the textbook diagram.

tanθB=nb

na

Geometric Optics /Spherical Mirrors

f =R/21f=

1d i

+1do

m=hi

ho

=−d i

do

Refraction with Spherical Boundary

nair

do

+nglass

d i

=nair−nglass

r curvature

f =nglass rcurvature

nglass−nair

m=−nair d o

nglass d i

Refraction at a planeLateral Magnification is 1.

nair

do

=−nglass

d i

Lens-maker's Equation1f=(n−1)( 1

R1

−1R2

)n = index of refractionR's = radii of curvature

InterferenceIn Young's double-slit experiment, only the path length differs. D is space between holesPath Length DifferenceΔ L=d sinθ

Phase Differenceϕ=(d sinθ)⋅(2π)/λ

Constructive interference at

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

ϕ=2π(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 IntensityIo = intensity of each source

I=4Io cos2(12ϕ)

DiffractionAny pair of rays seperated by a/2 has the same phase difference. “a” is width of holeDark fringes at

a sinθ=mλ ,m=±1,±2

Single Slit diffraction intensity

I (θ)=I m( sinαα )

2, Im is max intensity

α=1/2ϕ=παλ

sinθ

Circular AperaturesLocation of first dark fringe

sin θ1=1.22 λDiameter

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

θR=1.22 λD

Interference Intensity for Two “Wide” Slits

I=I mcos2(ϕ

2)( sinα

α )2