Useful Math Equations for AP Physics
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Law of Cosines
Volume of Sphere
Volume of Cylinder
Area of a Circle
Volume of Right Cone
Pythagorean Theorem
Celsius to Fahrenheit
Fahrenheit to Celsius
Celsius to Kelvin
c2=a2 + b2 – 2abcosC
V=(4/3) πr3
V=πr2h
A=πr2
V=(1/3) πr2h
a2+b2=c2
F=(9/5)C+32
C=(F-32)(5/9)
K= C+273
Unit 1 Vectors and One-Dimensional Equations
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Average Speed
Average Acceleration
Velocity Equation
Displacement Equation
Displacement Equation
Velocity Squared Equation
Initial Velocity in Y axis
Initial Velocity in X axis
Instantaneous Velocity
v=Δx/Δt
a=Δv/Δt
v=vo + at
Δx= vot + 1/2at2
x=1/2(vo+v)t
v2=vo2 + 2ax
voy=vosinθ
vox=vocosθ
dy = Δx
dx Δt
9
Instantaneous Acceleration
d2y = Δx OR dy = Δv
dx2 Δt
dx Δt
Unit 2 Multi-Dimensional Motion Equations
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Radian Angle
Average Angular Velocity
Angular Acceleration
Angular Velocity
Angular Displacement
Angular Displacement
Angular Velocity Squared
Linear and Angular Speed
Tangential Acceleration
Centripetal Acceleration
θ=s/r OR 1 rad=57.3˚ OR 2πrad=360˚
ω=Δθ/Δt
α=Δω/Δt
ω= ωo + αt
θ=1/2(ω + ω0)t
θ=ωot + 1/2 αt2
ω2= ωo2 + 2α Δθ
v=r ω
at=rα
ac=v2/r OR ac=r ω2
11 Tangential Speed
vt=2πr/T
Unit 3 Newton’s Laws and Uniform Circular Motion Equations
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Newton’s Second Law
Static Frictional Force
Kinetic Frictional Force
Gravitational Force
Weight
Apparent Weight
Force of Gravity in Y axis
Force of Gravity in X axis
Coefficient of Friction
Centripetal Force
Velocity when you have no idea
Velocity of Orbiting Bodies
Period of a Satellite
Kepler’s Law of Periods
ΣF=ma
fs = μsFN
fk = μkFN
F=Gm1m2/r2
W=mg
Wa=mg + ma
Fgy=mgcosθ
Fgx=mgsinθ
μ=tanθ
Fc=mv2/r
v=(gr)1/2
v=(GMe/r)1/2
T=2πr3/2/(GMe)1/2
r3/T2=GM/4π2 OR T2/r3=T2/r3
Unit 4 Energy, Momentum, and Special Relativity Equations
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Work done by Constant Force
Kinetic Energy
Potential Energy
Spring Energy
Hooke’s Law of Springs
Law of Conservation of Energy
Power
Momentum
Impulse and Momentum
Conservation of Linear Momentum
Center of Mass
Center of Velocity
Coefficient of Restitution
Time Dilation
Length Contraction
Relativistic Momentum
Einstein’s Combined Momentum Equation
Relativistic Relative Velocity
Einstein’s Relation between Mass and Energy
Dot Product
W=(Fcosθ)s
KE=1/2mv2
PE=mgh=mgvt
SE=1/2kx2
Fs=-kx
WNC=ΔKE + ΔPE + ΔSE
P=work/time OR P=Fv
p=mv
FΔt=mΔv
(m1v1 + m2v2)o = (m1v1 + m2v2)f
xcm= m1x1 + m2x2/m1 + m2
vcm= m1v1 + m2v2/m1 + m2
e= vf2 – vf1/ vo1 – vo2
t=to γ
L=Lo/ γ
p=mv γ
E2= p2c2 + m2c4
VAB= VAC + VCB/1+ VAVB/c2
E=mc2 γ
(ax)(bx)+(ay)(by) OR |a||b|cosθ
Unit 5 Circular Dynamics and Simple Harmonic Motion Equations
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Torque
Lever Arm
Cross Product
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Newton’s Second Law for Rotating Objects
Moment of Inertia of a Body
Inertia of hoop/hollowed cylinder
Inertia of cylinder/disk
Inertia of Thin Rod, axis through center
Inertia of Thin Rod, axis through one end
Inertia of Solid Sphere, axis through center
Inertia of Solid Sphere, axis tangent to surface
Inertia of Thin-Walled Sphere, axis through center
Inertia of Thin Rectangular Sheet, axis through center
Inertia of Thin Rectangular Sheet, axis through edge
Work by Torque
Power by Torque
Rotational Kinetic Energy
Impulse and Momentum with Torque
SHM Position
SHM Velocity
SHM Acceleration
Angular Frequency
Conservation of Energy
τ=Fl
l=r(sinθ)
cx=aybz-azby OR |a||b|sinθ
cy=azbx-axbz
cz=axby-aybx
OR use matrix determinant
Σ τ=I α
I=Σmr2
I=MR2
I=(1/2)MR2
I=(1/12)ML2
I=(1/3)ML2
I=(2/5)MR2
I=(7/5)MR2
I=(2/3)MR2
I=(1/12)ML2
I=(1/3)ML2
W=τθ=Frθ=I αθ
P=τω=τθ/t
KErot=(1/2)Iω2
τ Δt/ ΔL
x=Acos(ωt)
v=-Aωsin(ωt)=Aω
a=-Aω2cos(ωt)=- ω2x=Aω2
ω=2πf=(k/m)1/2
E=(1/2)mv2+ (1/2)Iω2+
mgh+(1/2)kx2
Unit 6 Fluids, Thermodynamics Equations
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Density
Pressure
Pressure and Depth of Static Fluids
Height using Pressures
Archimedes Principle
Buoyant Force when fully submerged
Bernoulli’s Equation
ρ=m/V
P=F/A
P=P0+ ρgh
h1=ΔP/( ρg)= (ρ2h2)/ ρ1
Fb=Wfluid
Fb/W= ρfVg/ρsVg
P1+(1/2)ρv12+ ρgh1=
P2+(1/2)ρv22+ ρgh2
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Mass Flow Rate
Volume Flow Rate
Length Expansion due to heat
Specific Heat Capacity
Heat in a closed system
Heat supplied/removed in phase change
Temperature of System derivation
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Latent Heat
Conduction
Radiation
Ideal Gas Law
Root Mean Square Velocity
Internal energy of monatomic gas
Kinetic Molecular Theory
Boyle’s Law
Charles’ Law
Gay-Lussac’s Law
Number of molecules
Number of moles
First Law of Thermodynamics
Isobaric Process
Isochoric Process
Isothermal Process
Adiabatic Process
Adiabatic Ratio
Engine efficiency
Carnot engine efficiency
Relationship between heat and work for engines
Refrigerator efficiency
Entropy
M= ρ1A1v1= ρ2A2v2
Q=A1v1= A2v2
ΔL=L0αΔT
Q=mCpΔT
Qb+QL …=0
Q=mL
T=mACpATA+mBCpBTB/
mACA+mBCB
L=Q/m=cΔT
Q=kAt|ΔT|/L
Q=eσtAT4
PV=nRT=NkT
vRMS=(3RT/M)1/2=(3kT/m)1/2
U=(3/2)nRT
KE=(3/2)kT
V α 1/P
V αT
PαT
N=n(NA)
n=mass/molecular mass
U=ΔQ+ΔW
W=-PΔV
W=0
W=-nRTln(Vf/Vi)
W=-(3/2)nRT, Q=0
PiVi γ =PfVf γ, γ=Cp/Cv
e=|W|/|Qh|
e=1-(Tc/Th)
Qh=W+Qc
K=|Qc|/|W|
ΔS=(Q/T), W=TiΔS
Unit 7 Electrostatics Equations
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Coulomb’s Law
Electric Field
Electric Field of Point Charge
Electric Field of Parallel Plate Capacitor
Gauss’ Law
Electric Potential Energy and work
Electric Potential Difference
Electric Potential Difference for point charge
F=(k|q1||q2|)/r2
Ef=F/q0=-ΔV/Δs
E=k|q|/r2
E=q/(ε0A)= σ/ ε0
Φ=Σ(Ecosφ)ΔA=q/ ε0
AWB=EPE,A- EPE,B
V=EPE/q
V=kq/r=Ed
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Work for point charge in electric field
Charge on a plate
Dielectric constant
Capacitance
Electrical Potential Energy
Energy Density
W=qEd=qV
q=CV
κ=E0/E
C= (κε0A)/d
E=(1/2)qV=(1/2)CV2=q2/(2C)
Energy density= (1/2)κε0E2
Unit 8 Electric Circuits
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Ohm’s Law
Electric Current
Resistance
Resistivity affected by temperature
Resistance affected by temperature
Power
Resistance in Series
Resistance in Parallel
Kirchhoff’s Junction Rule
Kirchhoff’s Loop Rule
Capacitance in Series
Capacitance in Parallel
Capacitor Charging
Time Constant of RC circuit
Terminal Voltage
Internal Resistance
V=IR
I=Δq/Δt
R= ρL/A
ρ= ρ0[1+α(T-T0)]
R=R0[1+α(T-T0)]
P=IV=V2/R=I2R
Rs=R1+R2…
Rp=R1-1+R2-1…
I1=ΣI
ΣV=0
Cs=C1-1+C2-1…
Cp=C1+C2…
q=q0[1-e-t/(RC)]
τ=RC
VT=ξ-IR
r=( (ξ-V)/V)R
Unit 9 Magnetism
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Force on a moving charge in magnetic field
Radius of path in magnetic field
Mass of particle in mass spectrometer
Time for one revolution in magnetic field
Velocity inside mass spectrometer
Charge to mass ratio
Force on current in magnetic field
Torque on Current-carrying wire
Magnetic field on infinitely long, straight wire
Magnetic field on circular, toroidal loop
Magnetic field on solenoid
Motional EMF
F=qvsinθB
r=(mv)/(qB)
m=((er2)/(2V))B2
t=2πm/(qB)
v=E/B
q/m=v/(rB)
F=Il Bsinθ
τ=NIABsinθ
B=(μ0I)/(2πr)
B=(Nμ0I)/(2R)
B= μ0nI= (μ0IN)/L
ξ=vBL
13 Magnetic Flux
14 EMF in a coil
Φ=BAcosθ= -ξΔt
ξ=-NΔΦ/Δt
Unit 10, 11 Waves, Sound, and Optics
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Velocity of a wave
Frequency
Speed of a wave on a string
Distance moved per second of string particle
Position and Time of wave in +x direction
Position and Time of wave in -x direction
Sound Intensity
Decibel Level (intensity level of sound)
Source moving toward stationary observer
Source moving away from stationary observer
Observer moving toward stationary source
Observer moving away from stationary source
Difference in frequencies with Doppler Effect
Diffraction of a wave
Diffraction of a wave, through circular opening
Transverse Standing Waves (strings)
17 Longitudinal Standing Waves (gases)
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Concave mirror radius
Convex mirror radius
Mirror equation
Magnification of mirror
Index of Refraction
Snell’s Law
Apparent Depth
Concave Mirror
Convex Mirror
Concave Lens
Convex Lens
v=fλ
f=1/T
v=[F/(m/l)]^(1/2)
d=4Af
y=Asin(2πft-(2πx)/ λ)
y=Asin(2πft+(2πx)/ λ)
I=P/A=P/4πr2
β=10log(I/I0)
f0=fs(1/(1-(vs/v))
f0=fs(1/(1+(vs/v))
f0=fs(1+(v0/v))
f0=fs(1-(v0/v))
Δf=2fs(v0/v)
sinθ=λ/D
sinθ=1.22λ/D
fn=nv/2L, n is any positive
integer
fn=nv/4L, n is an odd positive
number if only one open end
of tube
f=R/2
f=-R/2
do-1+di-1-1=f-1
m=-di/do=hi/ho
n=c/v
n1sinθ1=n2sinθ2
d’=dn2/n1
f=(+), d=(+) front, (-) behind
f=(-), d=(-)
f=(-), d=(-)
f=(+), d=(+) front, (-) behind
Unit 12 Modern Physics, Nuclear Physics, Quantum Physics
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Speed of Light in Vacuum
c=(1/ε0μ0)1/2
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Doppler Effect on Electromagnetic Waves, together
Doppler Effect on Electromagnetic Waves, apart
Malus’ Law with Polarization
Diffraction (Young’s Double Slit Experiment formula)
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Thin Film Interference
Wave Particle Duality
Energy of a photon
Conservation of energy with photons
Momentum of a photon
The Compton Effect
De Broglie Wavelength
Heisenberg Uncertainty Principle
14 Atomic Mass Number
15 Radius of a nucleus
16 Radioactive Decay
f0=fs(1+(vrel/c))
f0=fs(1-(vrel/c))
S=S0cos2θ
sinθ=mλ/d (change to m+1/2
for dark light)
λfilm=λvacuum/n
E=nhf
E=hf
hf=KE+W0
p=h/λ
λ’-λ=h(1-cosθ)/mc
λ=h/p
(Δpy)( Δy)≥h/4π,
(ΔE)(Δt)≥h/4π,
A=Z+N
r=(1.2x10-15)A1/3
A=A0e-kt