PHYSICS 221 Exam 3 & Final Spring 2012 Formula/Information...
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Transcript of PHYSICS 221 Exam 3 & Final Spring 2012 Formula/Information...
PHYSICS 221 Exam 3 & Final Spring 2012Formula/Information Sheet
• Basic constants:
Gravitational acceleration g = 9.8 m/sec2
Permittivity of free space ε0 = 8.8542× 10−12 C2/N·m2 [ k = 1/4πε0 = 8.9875× 109 N·m2/C2]Permeability of free space µ0 = 4π × 10−7 T·m/A [ km = µ0/4π = 10−7 Wb/A·m]Elementary charge e = 1.60× 10−19 CUnit of energy: electron volt 1 eV = 1.60× 10−19 JUnit of energy: kilowatt-hour 1 kWh = 3.6× 106 JPlanck’s Constant h = 6.626× 10−34 J sec
• Properties of some particles:Particle Mass [kg] Charge [C]
Proton 1.67× 10−27 +1.60× 10−19
Electron 9.11× 10−31 −1.60× 10−19
Neutron 1.67× 10−27 0
• Some indefinite integrals: ∫dxx
= lnx∫
dxa+bx
= 1b
ln (a+ bx)∫dx
(x2+a2)3/2= x
a2√x2+a2
∫x dx
(x2+a2)3/2= − 1√
x2+a2∫dx√x2±a2
= ln (x+√x2 ± a2)
∫x dx√x2±a2
=√x2 ± a2
• Equations for Thermodynamics:
Thermal Expansion ∆L = αLo∆T∆V = βVo∆T
Thermal Stress FA
= −Y α∆THeat Q = mc∆T
Q = nC∆TQ = ±mL
Heat Conduction H = dQdt
= kATH−TCL
H = AeσT 4
Hnet = Aeσ(T 4 − T 4S)
Stefan-Boltzmann Constant σ = 5.67× 10−8W/m2K4
Ideal Gas Law pV = nRT = NkTGas Constant R = 8.314J/molK
Ktr = 32nRT
vrms =√
3kTm
Mean Free Path λ = V
4π√
2r2N
Heat Capacity-monatomic gas CV = 32R
Heat Capacity-diatomic gas CV = 52R
Heat Capacity-monatomic solid CV = 3R
Molecular speeds (Maxwell Boltzmann) f(v) = 4π( m2πkT
)32 v2e−mv
2/2kT
Work W =∫ V2
V1pdV
First law of Thermodynamics ∆U = Q−WAdiabatic process Q = 0Isochoric process W = 0Isobaric process W = p(V2 − V1)Isothermal process ∆T = 0Ideal Gases CP = CV +R
γ = CPCV
Efficiency of Heat Engine e = WQH
= 1 + QCQH
The Otto Cycle e = 1− 1rγ−1
The Carnot Cycle e = 1− TCTH
Entropy ∆S =∫ 2
1
dQT
S = k ln(w)
• Equations for Periodic Motion and Waves:
f = 1T
ω = 2πfHook’s Law Fx = −kx
ω =√
km
General solution to SHM x(t) = A cos(ωt+ φ)Total Energy in SHM E(t) = 1/2mv2
x + 1/2kx2
Torsional Pendulum ω =√
κI
Simple Pendulum ω =√
gl
Physical Pendulum ω =
√mgdI
Damped Harmonic Motion Solution x(t) = Ae−b/2m)t cos(ω′t)
ω′ =√
(k/m− b2/4m2)
A = Fmax√(k−mω2
d)2+b2ω2
d
Wave velocity v = λf
Wave Equation ∂2y(x,t)
∂x2= 1
v2∂2y(x,t)
∂t2
General solution to the wave equation y(x, t) = A cos(ω(x/v − t))y(x, t) = A cos(kx− ωt)
Velocity of a wave on a string v =√
Fµ
Average Power in a wave Pave = 1/2√µFω2A2
Inverse Square Law of Intensity I1I2
=r22r21
For strings fixed at both ends fn = n(v/2L)(n = 1, 2, 3, ..)
f1 = 12L
√Fµ
Pressure in a sound wave pmax = BkA
Longitudinal wave in a fluid v =√
Bρ
Longitudinal wave in an ideal gas v =
√γRTM
Longitudinal wave in a solid rod v =√
Yρ
Sound Intensity I = 1/2√ρBω2A2
=p2max2ρv
Definition of Intensity Levels β = (10dB) log II0
reference sound intensity I0 = 10−12W/m2
Standing sound waves, open pipe fn = nv2L
(n = 1, 2, 3..)Standing sound waves, stopped pipe fn = nv
4L(n = 1, 3, 5, ...)
Beat Frequency fbeat = fa − fbDoppler Effect fL = v+vl
v+vsfs
• Basic Equations for Waves, Interference and Diffraction:
Wave Equation ∂2f(x,t)
∂x2= 1
v2∂2f(x,t)
∂t2
Plane EM wave traveling in the +x direction E(x, t) = Em sin(kx− ωt)B(x, t) = Bm sin(kx− ωt)
Speed of an EM wave [m/s] c = 1√µ0ε0
= EmBm
= E(x,t)B(x,t)
Wave length of an EM wave [m] λ = cf
Wave number of an EM wave k = 2πλ
Poynting vector [J/s·m2] ~S = 1µ0
~E × ~B
Time-averaged S [J/s·m2] Save = EmBm2µ0
Intensity of an EM wave [J/s·m2] I = SaveTotal energy of an EM wave [J] U = I A tTotal momentum of an EM wave |~p| = U
c
Law of Reflection θincident = θreflected
Snell’s Law n1 sin(θ1) = n2 sin(θ2)Law of Malus I = Imax cos2 φBrewster’s Law tan θp = nb
na
Mirror Equation 1f
= 1s
+ 1s′
Refraction from a Spherical Surface nb−naR
= nas
+ nbs′
Lens Equation 1f
= 1s
+ 1s′
Lens Maker’s Equation 1f
= (n− 1)(
1R1− 1
R2
)Magnification M = hi
ho= −s′
s
Double Slit Constructive Int. d sin(θ) = mλDouble Slit Destructive Int. d sin(θ) = (m+ 1
2)λ
Intensity Maxima Iθ = Io cos2(φ/2)φ = 2πd
λsin(θ)
Single Slit Diffraction Dest. Int. sin(θ) = mλa
Single Slit Diffraction Intensity I = Io
[sin(πa(sin θ)/λ)πa(sin θ)/λ
]2
Multiple Slit Diffraction Const. Int. d sin(θ) = mλ
Two Slit Diffraction Intensity I = Io cos2 φ2
[sin(β/2)β/2
]2
φ = 2πdλ
sin(θ)β = 2πa
λsin(θ)
Diffraction Grating d sin(θ) = mλResolving Power R = λ
∆λ= Nm
X-Ray Diffraction 2d sin(θ) = mλResolving Power of Circular Apertures sin(θ1) = 1.22 λ
D