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# FORMULAS

## KINEMATICS

 Non-Uniform Motion d v1 v2 t a $\Delta \vec{d} = \dfrac{\vec{v}_1 + \vec{v}_2}{2} \Delta t$ ✓ ✓ ✓ ✓ ✘ $\vec{v}_2^2 = \vec{v}_1^2 + 2a \Delta\vec{d}$ ✓ ✓ ✓ ✘ ✓ $\Delta \vec{d} = \vec{v}_1 \Delta t + \frac{1}{2}\vec{a} \Delta t^2$ ✓ ✓ ✘ ✓ ✓ $\Delta \vec{d} = \vec{v}_2 \Delta t - \frac{1}{2}\vec{a} \Delta t^2$ ✓ ✘ ✓ ✓ ✓ $\vec{v}_2 = \vec{v}_1 + \vec{a} \Delta t$ ✘ ✓ ✓ ✓ ✓
Acceleration due to gravity
$g = -9.81 m/s^2$
Uniform Motion
$d = v\,t$
Components:
$\vec{v}_y = \vec{v}\sin\theta \\ \\ \vec{v}_x = \vec{v}\cos\theta$
For $\theta$ = angle of elevation. Average: Distance uses average speed if the 2 trips have the same time.

## FORCES

Newton's 1st Law Objects at rest or in constant motion remain that way unless a net force acts on it. Newton's 2nd Law Net force is directly proportional to the product of acceleration and mass (F = m × a). Newton's 3rd Law For every action, there is an equal and opposite reaction.
Weight = Force of Gravity
$F_g = m × g$
Friction
$F_{kinetic\ friction} = \mu_k × F_N$
$F_{static\ friction} = \mu_s × F_N$
Circular Motion
$a_c = \dfrac{v^2}{r}$
$F_c = m \dfrac{v^2}{r}$
v is tangential velocity

Fc points to circle center

## WORK, ENERGY, POWER

$\text{Power} = \dfrac{Energy}{Time} = \dfrac{\Delta W}{\Delta t} = F × \vec{v}$
$\%\ \text{Efficiency} = \dfrac{Energy\ used}{Energy\ in} × 100 \%$
$E_{total} = KE_{total} + PE_{total}$
$\Delta E = KE_2 - KE_1 + PE_2 - PE_1$
$W = \Delta E$
$W = F × d × \cos\theta$
$KE = \frac{1}{2}mv^2$
$PE = m × g × h$

## MOMENTUM

$J = \Delta p$
p = momentum
J = impulse
$p = m × \vec{v}$
$J = F × \Delta t$
$\tau = r × F × \sin\theta$
τ = torque
Elastic
$KE_1 = KE_2$
$p_1 = p_2$
Inelastic
$p_1 = p_2$
$m_{a\,i}v_{a\,i} + m_{b\,i}v_{b\,i} = m_{a\,f}v_{a\,f} + m_{b\,f}v_{b\,f}$
$\frac{1}{2}m_{a\,i}v_{a\,i}^2 + \frac{1}{2}m_{b\,i}v_{b\,i}^2 = \frac{1}{2}m_{a\,f}v_{a\,f}^2 + \frac{1}{2}m_{b\,f}v_{b\,f}^2$

## GRAVITY

$F_g = \dfrac{Gm_1m_2}{r^2}$
G = universal gravitational constant
$g\ field = \dfrac{-GM}{r^2} = \dfrac{F}{m}$
$V\ potential = \dfrac{-GM}{r}$
$U\ potential\ energy = \dfrac{-GMm}{r}$
$U = V × m$
Kepler's 3rd
$\dfrac{R^3}{T^2} = \dfrac{GM}{4\pi^2}$
T = period
M = mass of central body

Constants:
$G = 6.67 × 10^{-11} \frac{Nm^2}{kg^2}$
$r_{Earth} = 6.4 × 10^{6} m$
$m_{Earth} = 5.98 × 10^{24} kg$
$m_{Sun} = 2 × 10^{30} kg$
$m_{Moon} = 7.34 × 10^{22} kg$

## LIGHT

$v = f × λ$
$\dfrac{1}{f} = \dfrac{1}{i} + \dfrac{1}{o} = \dfrac{2}{r}$
f = focal length
i = image distance
o = object distance
$M = \dfrac{h_i}{h_o} = -\dfrac{i}{o}$
+M = magnified
-M = smaller
|M| > 1 = upright
|M| < 1 = inverted
hi = image height
ho = object height
$n = \dfrac{c}{v}$
n = index of refraction
c = speed of light in vacuum
v = speed in other medium
$c = 3.00 × 10^{8} \tfrac{m}{s}$
$n_1 \sin\theta_1 = n_2 \sin\theta_2$
\begin{align} Intensity & = \dfrac{Power}{Area} \\ \\ & = \dfrac{Energy}{(Area)(time)} \end{align}
Constructive
$m\,λ = d\,\sin\theta$
d = width of slits
m = fringe order integer
λ = wavelength
Destructive
$\left(m + \frac{1}{2}\right)\,λ = d\,\sin\theta$
Small Angle Approximation
$m\,λ = \dfrac{d\,y}{x}$
y = fringe distance on screen
x = screen distance from slits
Thin Films
$2d = \dfrac{λ}{n}$
n = index of refraction
$\left(m + \frac{1}{2}\right)\,λ = 2\,d\,n$
(Constructive)
$m\,λ = 2\,d\,n$
(Destructive)
$\left(m + \frac{1}{4}\right)\,λ = 2\,d\,n$
(Destructive,
antireflective)
Polarization
$I = I_0 \,\cos^2\theta$
Brewster
$\tan\theta = \dfrac{n_2}{n_1}$
Resolution
$\theta = \dfrac{1.22\,λ}{a}$
a = aperature

## SOUND

$\vec{v}_{sound} = 331 \tfrac{m}{s}$
$v = f × λ$
$v = 331 + 0.6 × T$
speed of sound changes with air temp.
v in m/s, T in ˚C
\begin{align} Intensity & = \dfrac{Power}{Area} \\ \\ & = (pressure)(\vec{v}) \end{align}
Intensity in W/m2
Surface area sphere = $4\pi r^2$
$dB = 10\,\log\left(\dfrac{I}{I_0}\right)$
I0 is threshold of human hearing
I0 = 10-12 W/m2
$\text{times louder} = 10 ^\left(\tfrac{dB_H \,-\, dB_L}{10}\right)$
dbH is higher
dbL is lower
Strings & Open Pipes
$λ = \dfrac{2L}{n}$
n = 1, 2, 3, 4, ...
$f = \dfrac{n\,v}{2L}$
Closed Pipes
$λ = \dfrac{4L}{n}$
n = 1, 3, 5, 7, ...
$f = \dfrac{n\,v}{4L}$
Frequency, Doppler
$f_{beat} = \Big|f_1 - f_2\Big|$
$f’ = f × \dfrac{\left(v \pm v_D\right)}{\left(v \mp v_s\right)}$
together: ±
apart: ∓

## SIMPLE HARMONIC MOTION

Strings/Springs
$\vec{v} = \sqrt{\dfrac{F_T}{\mu}}$
$\mu$ = linear density (kg/m)
FT = Tension (N)
L = length (m)
m = mass (kg)
v = speed (m/s)
$\mu = \dfrac{m}{L}$
All Waves
$\vec{v} = f × λ$
$f = \dfrac{1}{T}$
$E_{T} = KE + PE$
$ω = 2 \pi f$
Springs
$T = 2\pi \sqrt{\dfrac{m}{k}}$
T = period
m = mass
k = spring const.
$F = -k × x$
$KE = \frac{1}{2}m\,v^2$
$PE = \frac{1}{2}k\,x^2$
$ω^2 = \dfrac{k}{m}$
Pendulums
$T = 2\pi \sqrt{\dfrac{L}{g}}$
T = period
L = length
g = gravity
$KE = \frac{1}{2}m\,v^2$
$PE = m\,g\,h$
$ω^2 = \dfrac{g}{L}$
Wave Equations
$KE = \frac{1}{2} mω^2(x_0^2 - x^2)$
$PE = \frac{1}{2} mω^2(x^2)$
$E_{T} = \frac{1}{2} mω^2(x_0^2)$
$\vec{x}_{max} = A$
$\vec{v}_{max} = -ω \, A$
$\vec{a}_{max} = -ω^2 \, A$
\begin{align} v & = ω \sqrt{A^2 - x^2} \\ & = \pm \sqrt{\frac{k}{m} (A^2 - x^2) } \\ & = \pm \vec{v}_{max} \sqrt{1 - \frac{x^2}{A^2}} \end{align}
At xmin (x0 = 0)
$\vec{x} = A \sin (ω\,t)$
$\vec{v} = -ω \, A \cos (ω\,t)$
$\vec{a} = -ω^2 \, A \sin (ω \,t)$
At xmax (x0 = A)
$\vec{x} = A \cos (ω\,t)$
$\vec{v} = -ω \, A \sin (ω\,t)$
$\vec{a} = -ω^2 \, A \cos (ω \,t)$
v = velocity
a = acceleration
x = displacement
A = amplitude
ω = angular frequency

## ELECTRIC & MAGNETIC FIELDS

$F_{moving\ charge} = q\,v\,B\,\sin\theta$
$F_{current\ wire} = I\,L\,B\,\sin\theta$
$F = \dfrac{k\,q_1\,q_2}{r^2} = \dfrac{q_1\,q_2}{4\,\pi\,ε_0\,r^2}$
$k = 8.99 × 10^9 \, \tfrac{Nm^2}{C^2}$
$φ = N\,B\,A\,\cos\theta$
φ = flux
$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} + \dfrac{I_s}{I_p}$
V = voltage
N = number of turns
I = current
s = secondary
p = primary
$U = q\,V$
U = electric potential energy
V = electric potential
$V = E\,d$
d = separation distance
E = electric field
$V = \dfrac{\Delta\,U}{q} = \dfrac{k\,q}{r}$
$q = N × e × F_k$
N = # turns
$ε = \dfrac{\Delta\,p}{\Delta\,t}$
$ε = B\,L\,v$
ε = EMF, Volts
B = magnetic field
L = length
v = speed
\begin{align} U & = q\,\Delta d \\ \\ & = q\,E\,d \\ \\ & = \dfrac{k\,q_1\,q_2}{r} \end{align}
$E = \dfrac{F}{q} = \dfrac{k\,q}{r}$
Constants:
$q_{e^-} = -1.0602 × 10^{-19} C$
$q_{p^+} = +1.0602 × 10^{-19} C$
$m_{electron} = 9.11 × 10^{-31} kg$
$m_{proton} = 1.67 × 10^{-27} kg$
$Faraday\ k = 6.02 × 10^{23} \tfrac{C}{mol\,e^-}$
Magnetism
$B = µ_0\,i\,n$
$µ_0 = 4\,\pi × 10^{-7} \tfrac{T\,m}{A}$
i = current
Straight Wire
$B = \dfrac{µ_0\,i}{2\,\pi\,r}$
Center of Loop
$B = \dfrac{µ_0\,i\,n}{2\,r}$
Solenoid Turns
$B = \dfrac{µ_0\,i\,n}{L}$

## DC & AC CIRCUITS

$i = \dfrac{\Delta\,q}{\Delta\,t}$
$V = i \, r$
$C = \dfrac{Q}{V}$
$R = \dfrac{p\,I}{A}$
\begin{align} P & = i\,V \\ \\ & = \dfrac{V^2}{R} \\ \\ & = i^2\,R \end{align}
$C = \dfrac{k\,E_0\,A}{d}$
k = constant
A = area
d = distance
$E_0 = 8.85 × 10^{-12} \tfrac{C^2}{m^2\,N}$
\begin{align} E & = \frac{1}{2}Q\,V \\ \\ & = \frac{1}{2}C\,V^2 \\ \\ & = \frac{1}{2}\frac{Q^2}{C} \end{align}
$Power = \dfrac{Energy}{time}$
Series Circuit
$R_T = R_1 + R_2 + R_3 + \, ...$
$V_T = V_1 + V_2 + V_3 + \, ...$
$i_T = i_1 = i_2 = i_3 = \, ...$
$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + \, ...$
Parallel Circuit
$\dfrac{1}{R_T} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + \, ...$
$V_T = V_1 = V_2 = V_3 = \, ...$
$i_T = i_1 + i_2 + i_3 + \, ...$
$C_{eq} = C_1 + C_2 + C_3 + \, ...$

## THERMODYNAMICS

1st Law Thermodynamics
$\Delta U = Q - W$
+Q = heat added to system
-Q = removed from system
+W = work done by system (out)
-W = work done on system (in)
$W = Q_{in} - Q_{out}$
Specific Heat
$C = m × c$
c = specific heat = specific heat capacity
C = heat capacity
Q = heat energy
$Q = m × c × \Delta t$
$Q = m × L$
(for phase changes)
L = latent heat
\begin{align} \text{molar heat capacity} & = c × molar\ mass \\ & = \dfrac{Q}{n × \Delta t} \end{align}
Carnot Heat Efficiency
$Efficiency_{carnot} = \left(1 - \dfrac{T_L}{T_H}\right) × 100\%$
$Work = Q_H - Q_L$
H = hot, in
L = cold, out

## QUANTUM

$E = h × f$
$c = f × λ$
$E = m × c^2$
$\frac{1}{2}mv^2 = qV$
$1 eV = 1.602 × 10^{-19} J$
$h = 6.63 × 10^{-34} Js$
h = Planck's constant
$KE = hf - φ$
$φ = h × f_0$
$hf = φ - qV$
$λ = \dfrac{h \, c}{E} = \dfrac{h}{p} = \dfrac{h}{m \, v}$
Relativity
$L = L_O \sqrt{ 1 - \dfrac{v^2}{c^2} }$
$t = \dfrac{t_0}{ \sqrt{ 1 - \dfrac{v^2}{c^2} } }$

# CONSTANTS

 Quantity Symbol Value Speed of light in a vacuum c 2.9979 × 108 m/s (exact) Planck's constant h 6.6260 × 10–34 J·s Universal gas constant R 0.08205 L·atm/(mol·K) 8.3145 J/(mol·K) 8.3145 m3·Pa/(mol·K) 8.3145 L·kPa/(mol·K) 0.083145 L·bar/(mol·K) 62.364 L·Torr/(mol·K) Avogadro constant NA 6.0221 × 1023 formula units/mol Mass of an electron me 9.10938 × 10–28 g 5.4858 × 10–4 amu Mass of a proton mp 1.67262 × 10−24 g 1.00727 amu Mass of a neutron mn 1.674927 × 10−24 g 1.00866 amu Charge on an electron e –1.60217 × 10–19 C Boltzmann constant k 1.3806 × 10–23 J/K Faraday constant F 96485 C/mol of electrons Rydberg constant RH 1.09737 × 107 m–1 2.179872 × 10–18 J 3.289842 × 1015 s–1 Electric constant ε0 8.85418 × 10–12 F/m 8.85418 × 10–12 C2/(J·m)

# CONVERSIONS

 Length 1 in = 2.54 cm (exact) 1 Å = 1 × 10–10 m 1 mi = 5280 ft 1 mi = 1609.3 m 1 ft = 0.3048 m 1 m = 1.094 yd 1 km = 0.621 miles(mi) Volume 1 mL = 1 cm3 1 L = 1.0567 qt 1 gal = 3.7854 L 1 m3 = 1000 L Mass 1 lb = 453.6 g 1 g = 6.022 × 1023 amu 1 kg = 2.20 lb 1 lb = 16 oz 1 oz = 28.35 g 1 ton = 2000 lb 1 metric ton = 1 Mg Energy 1 cal = 4.184 J 1 L·atm = 101.325 J 1 eV = 1.602 × 10–19 J 1 J = 1 kg·m2/s21 J = 1 m3·Pa 1 J = 1 L·kPa 1 J = 1 C·V Pressure 1 atm = 760 Torr 1 atm = 760 mmHg 1 atm = 101325 Pa 1 atm = 14.7 psi 1 atm = 1.01325 bar 1 bar = 100 kPa 1 Torr = 1 mmHg Temperature TK = TC + 273.15 TC = (5/9)(TF – 32) TF = 32 + (9/5)TC

# MAGNITUDES

#### Symbol

1015 peta P
1012 tera T
109 giga G
106 mega M
103 kilo k
102 hecto h
101 deca da
10−1 deci d
10−2 centi c
10−3 milli m
10−6 micro µ
10−9 nano n
10−12 pico p
10−15 femto f
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