PHYS 15200 Formulas and Constants
Constants
g = 9.81 m/s2
G = 6.67´10-11 N-m2/kg2
ρ water = 1000 kg/m 3
Patm = 1.013 × 105 Pa
Kinematics
  

r = r0 + v0t + 12 at 2
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Momentum and Impulse
𝑝⃗ = 𝑚𝑣⃗
Δ𝑝⃗ = 0 if no external forces act
𝐽⃗ = ∫ 𝐹⃗%() 𝑑𝑡 = Δ𝑝⃗
d 2x
+ ω 2x = 0
2
dt
x(t) = Acos(ω t + φ )
ω = k /m (mass-spring)
Center of Mass
ω = g/L (simple pendulum)


RCM = Σmi ri /M


VCM = Σmi vi /M


ACM = Σmi ai /M


P = Σmi vi



ΣFext = MACM = dP/dt
Rotational Kinematics
s = Rθ , v = Rω , atan = Rα
Uniform Circular Motion
θ = θ 0 + ω 0t + 12 α t 2
Dynamics
Simple Harmonic Motion
 
Δp = Favg Δt
!
!
!
!
| v2,i − v1,i |=| v2, f − v1, f |
v 2 = v02 + 2a(x − x0 )



vobj, A = vobj, B + v B, A
ac = v 2 /r = ω 2 r
v = ωr
ω = 2π /T = 2π f


Στ ext = dLtotal /dt
ω 2 = ω 02 + 2α (θ − θ 0 )
Rotational Dynamics
8⃗*+, = 𝑀𝑎⃗-.
∑F
I points = Σmi ri2 , I = ∫ r 2 dm
ω = κ /I (torsion pend.)
ω=
MgRcm /I (physical pend.)
𝜔
1
=
2𝜋 𝑇
Fluids
𝑃 = 𝐹/0& /𝐴
𝑓=
P2 = P1 + ρ g(h1 − h2 )
FB = ρ fluidVdisp g
A1v1 = A2 v2
P + 12 ρv 2 + ρ gh = const.
Harmonic Waves
∂2 y 1 ∂2 y
−
=0
∂x 2 v 2 ∂t 2
y(x,t) = Acos(kx  ω t)
k = 2π /λ , ω = 2π /T
v = f λ = ω /k


p = mv



Fnet = ma = dp/dt
I parallel = I CM + MD 2
Fgrav = mg
I CM ,rod = 121 ML2
Fgrav = Gm1m2 /r 2
I CM ,cylinder = 12 MR 2 (solid)
𝐼 ∝ 𝑃 ∝ 𝜔1 𝐴1
𝐹⃗!"#$%& = −𝑘𝑥
𝑓' = 𝜇' 𝑁, 𝑓! ≤ 𝜇! 𝑁
I CM ,cylinder = MR (hollow)
Waves on a String
Work and Kinetic Energy
I CM ,sphere = 25 MR 2 (solid)
  
τ =r×F


τ net = Iα
 
W = ∫ Fid 
K = 12 mv 2
Wtotal = ΔK
𝑃 = 𝐹⃗ ⋅ 𝑣⃗
Potential and Total Energy
U grav = mgy
U grav = −GMm/r
U spring = 12 kx 2
ΔE = ΔK + ΔU = Wnc
F = −dU (x)/dx


F = −∇U
2
I CM ,sphere = 23 MR 2 (hollow)
Rotational Energy
K rot = 12 I CM ω 2
K trans = MV
1
2
2
CM
K total = K trans + K rot
Wrot = ∫ τ dθ
Angular Momentum
  
L=r × p


L = Iω
Δ𝐿8⃗ = 0 if no external torques
v = FT /µ
µ = m/L
λn = 2L/n, n = 1,2,3,...
𝑓% = 𝑛𝑓2
Sound
I = Pav / A
β = (10 dB) log ( I / I 0 )
fr =
v ± ur
fs
v ± us
%0
𝑓% = 13 𝑛 = 1,2,3, …
%0
𝑓% = 43 𝑛 = 1,3,5, …
𝑓5(/) = 𝑓/ − 𝑓5