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PHY 211
v(t) = dx(t) dt
a(t) = dv(t) dt = d
(^2) x(t) dt^2
x(t) =
ˆ (^) t
0
v(t′)dt′
v(t) =
ˆ (^) t
0
a(t′)dt′
xf = xi + vit +^1 2 at^2 vf = vi + at vf^2 = vi^2 + 2a(xf − xi) vavg =^1 2 (vf + vi) ∆~r = ~vavg∆t F~net ≡
i
F^ ~i
F^ ~net = m~a Fg = w = mg | F~s| ≤ μs|~n| | F~k| = μs|~n| F^ ~S = −k~x
Y =
∆P
∆L/L 0
∆F
A∆L/L 0
S =
∆P
∆x/h
∆F
A∆x/ ω = vt r
ac = v^2 t r = rω^2
F^ ~G = −G m^1 m^2 r^2 ˆr
g = G mearth r^2 earth
W = F~ · d~ = | F~ ||d~| cos(θ)
W =
ˆ (^) xf
xi
Fxdx
W =
ˆ (^) Pf
Pi
F^ ~ · d~` =
ˆ (^) Pf
Pi
F‖d`
KE = K ≡
mv^2 Wnet = ∆K Wcons = −∆U = −∆PE PEg = Ug = mgh PEG = UG = −G m 1 m 2 r PES = US =
2 kx
2
~rcm ≡
∑^ i^ mi~ri i mi ~vcm ≡
∑^ i^ mi~vi i mi ~acm ≡
∑^ i^ mi~ai i mi ~p ≡ m~v
F~ = ∆~p ∆t ~p 1 ,f + ~p 2 ,f = ~p 1 ,i + ~p 2 ,i. J ≡~ F~ ∆t = ∆~p ~τ ≡ ~r × F~ r⊥ = r sin(θ) τ = r⊥F = Iα θ = s r ω = ∆θ ∆t
vt r α = ∆ω ∆t =^
at r ~v = −~r × ~ω θf = θi + ωit +
2 αt
2
ωf = ωi + αt ωf^2 = ω^2 i + 2α(θf − θi) ωavg =^1 2 (ωf + ωi) ~L = ~r × ~p = I~ω L = r(mv) sin θ
~τ =
∆L~
∆t W = τ ∆θ Krot =
Iω^2
ω = 2πf = 2 π T
ω =
k m
ω =
g `
ω =
mg`cm I x(t) = xmax sin(ωt) v(t) = ωxmax cos(ωt) a(t) = −ω^2 xmax sin(ωt)
E =
kx^2 max v = f λ μ ≡ m `
v =
F
μ
k ≡ 2 π λ ysw(x, t) =
[
A sin(kx)
]
sin(ωt) fstring = fopen−open = n
( (^) v 2 `
where n ∈ { 1 , 2 , 3 ,.. .}
` = n λn 2 where n ∈ { 1 , 2 , 3 ,.. .} fopen−closed = n
( (^) v 4 `
where n ∈ { 1 , 3 , 5 ,.. .} ` = n λn 4 where n ∈ { 1 , 3 , 5 ,.. .} ytw(x, t) = A sin(kx ∓ ωt)
fo = fs
v ± vo v ∓ vs
fbeat = |f 2 − f 1 |
I =
P
A
I =
P
4 πr^2 I 0 ≡ 1. 0 × 10 −^12
W
m^2
β[dB] = 10 log 10
I
I 0
P =
F
A
ρ = m V Fb = ρV g P = P 0 + ρgh
P + ρgy +
2 ρv
(^2) = const
Q = Φv ≡ ~v · A~ = vA cos(θ)
Q =
∆V
∆t A 1 v 1 = A 2 v 2
ρvA = ∆m ∆t ∆L = αL 0 ∆T L(∆T ) = L 0 (1 + α∆T ) ∆V = βv 0 ∆T
circumference of a circle C = 2πr area of a circle A = πr^2 surface area of a sphere A = 4πr^2
volume of a sphere V =
πr^3
If Ax^2 + Bx + C = 0, x =
−B ±
B^2 − 4 AC
2 A
loga(xy) = loga(x) + loga(y)
loga
x y
= loga(x) − loga(y)
loga (xy^ ) = y loga(x)
If ax^ = y, x = loga y = log 10 y log 10 a
ln y ln a If |θ| < 0 .5 radians, sin(θ) ≈ θ (in radians)
If |θ| < 0 .5 radians, tan(θ) ≈ θ (in radians) sin(−θ) = − sin(θ) cos(−θ) = cos(θ) sin(θA + θB ) = sin(θA) cos(θB ) + cos(θA) sin(θB ) cos(θA + θB ) = cos(θA) cos(θB ) − sin(θA) sin(θB )
sin(θA) sin(θB ) = cos(θA − θB ) − cos(θA + θB ) 2 cos(θA) cos(θB ) = cos(θA − θB ) + cos(θA + θB ) 2 sin(θA) cos(θB ) = sin(θA^ −^ θB^ ) + sin(θA^ +^ θB^ ) 2 Law of Cosines c^2 = a^2 + b^2 − 2 ab cos(C)
Law of Sines a sin(A) = b sin(B) = c sin(C)
x = r cos(θ) y = r sin(θ)
r =
x^2 + y^2
θ = tan−^1 (y/x) +
0 ◦, if x > 0 180 ◦, otherwise
If R~ = A~ + B, R~ x = Ax + Bx and Ry = Ay + By If R~ = A~ − B, R~ x = Ax − Bx and Ry = Ay − By
A^ ~ · B~ = B~ · A~ = AxBx + Ay By + Az Bz = | A~|| B~| cos(θ) | A~ × B~| = | A~|| B~||sin(θ)|
Newton’s constant G = 6. 67430 × 10 −^11 m^3 kg · s^2 speed of light c ≡ 2. 99792458 × 108 m/s elementary charge e = 1. 602176634 × 10 −^19 C
electrostatic constant k = 8. 987551792 × 109 N · m^2 C^2 vacuum permittivity 0 = 8. 854187813 × 10 −^12 F/m
vacuum permeability μ 0 = 1. 2566370621 × 10 −^6 N · A−^2 (1) ≈ 4 π× 10 −^7 N · A−^2 (2) Planck’s constant h = 6. 62607015 × 10 −^34 J · s ℏ ≡ h 2 π = 1. 054571817 × 10 −^34 J · s
standard gravity g = +9. 80665 m s^2 mass of earth mearth = 5. 9723 × 1024 kg mass of moon mmoon = 7. 346 × 1022 kg mass of sun msun = 1. 9885 × 1030 kg mass of electron me = 9. 1093837015 × 10 −^31 kg
mass of proton mp = 1. 67262192369 × 10 −^27 kg mass of neutron mn = 1. 67492749804 × 10 −^27 kg volumetric radius of earth rearth = 6. 371 × 106 m earth-moon distance rEM = 3. 844 × 108 m earth-sun distance rES = 1. 496 × 1011 m Density of air at sea level at 15◦^ C: ρ 0 = 1. 225 kg m^3 Earth’s total magnetic field strength at Huntington, WV: | B~earth| ≈ 5. 15 × 10 −^5 T Vertical component of Earth’s magnetic field strength at Hunt- ington, WV: Bearth,z ≈ 4. 70 × 10 −^5 T
Bohr radius aB ≡
ℏ^2
meke^2
= 5. 29177210903 × 10 −^11 m (4) Rydberg constant R ≡
4 πmea^2 B c
= 1. 0973731568160 × 107 m−^1 (6) hydrogen binding energy E 0 = 13.605693123 eV
PHY 213
q = N e F~ = k q^1 q^2 r^2 ˆr
E~ ≡ F~
q E~ = k q r^2 ˆr
E~ = k
dq r^2 ˆr
PE = U = k q 1 q 2 r V ≡ U q
= −W
q
Vf − Vi =
ˆ (^) Pf
Pi
E^ ~ · d~`
V = k q r
E~ = −
ˆı
∂V
∂x
∂V
∂y
∂V
∂z
ΦE ≡
E^ ~ · d A~
ΦE = E~ · A~ = A| E~| cos θ ΦE = qenclosed 0 C ≡ q V C = κ 0 A d 1 Cseries
C 1
C 2
Cparallel = C 1 + C 2 +...
U =
qV
U =
q^2 C U =
CV^2
u =
| E~|^2
I =
∆q ∆t I~ = nqA~vdrift q~v = I~`
V = IR
R = ρ` A ρ = ρ 0 (1 + α∆T ) R = R 0 (1 + α∆T ) P = IV P = V
2 R P = IR^2 V(t) = V 0 sin(ωt)
Pavg =
I 0 V 0
Pavg = IRMSVRMS Rseries = R 1 + R 2 +... 1 Rparallel
R 1
R 2
τ = RC I(t) = Imaxe−t/τ VR(t) = Vbatterye−t/τ VC = Vbattery
1 − e−t/τ^
VR(t) = Vmaxe−t/τ VC = −Vmaxe−t/τ F^ ~B = q~v × B~ F~ = I~` × B~ ~μ = N I Aˆ ~τ = ~μ × B~ U = −μ~ · B~
B = μ 0 I 2 πr B = N μ 0 I 2 r B = N μ^0 I = nμ 0 I F = μ^0 I^1 I^2 2 πr ΦB ≡
B^ ~ · d A~
ΦB = B~ · A~ = AB cos(θ)
EMF = − ∆Φtot ∆t
= −N ∆Φ^1
∆t EMF = N ABω sin(ωt)
VS
VP
NS
NP
IP
IS
EMF = −L
∆I
∆t U =^1 2
LI^2
u =
2 μ 0 B
2
τ =
L
R
XC = 1
2 πf C
ωC XL = 2πf L = ωL
IRMS =
VRMS
|Z|
|Ztotal| =
R^2 + (XL − XC )^2
tan φ =
XL − XC
R
ωres = 2πfres =
LC
Pavg = IRMSVRMS cos φ | E~| = c| B~|
S^ ~ = 1 μ 0
E^ ~ × B~
c =
√μ 0 0 v = f λ
Iavg = | E~max|| B~max| 2 μ 0 n ≡ c v n 1 sin(θ 1 ) = n 2 sin(θ 2 )
θc = sin−^1
n 2 n 1
P =
f =^
dO^ +
dI 1 f
nl − nm nm
r 1
r 2
m ≡ hI hO = − dI dO f = r 2 m = mobjective meyepiece
m ∼
N.P.
fe
` − fe do
M =
θtelescope θnaked eye
fo fe d sin(θ) = mλ where m ∈ { 0 , ± 1 , ± 2 ,.. .} d sin(θ) =
m −
λ
where m ∈ { 1 , 2 ,.. .} 2 t = mλ where m ∈ { 0 , 1 , 2 ,.. .} 2 t =
m −
λ
where m ∈ { 1 , 2 , 3 ,.. .} sin(θ) = 1. 22 λ D E = ∆mc^2 E = hf = pc
E = hf = ℏω = hc λ (∆x)(∆p) ≥
(∆t)(∆E) ≥
rn = n^2 Z aB
nλn = nh mevn = 2πrn
En = − Z
2 n^2 (13.86 eV)
1 λ
n^2 f
n^2 i
R
Q = (MA + MB − MC − MD )c^2 |~L| = mevrn = nℏ N (t) = N 0 e−λt^ = N 02 −t/t^1 /^2
λ = ln 2 t 1 / 2
R ≡ ∆N ∆t
R = N λ = N ln 2 t 1 / 2 r = rnucA^1 /^3
