1206 FINAL EXAM
Formula sheet - December 2009
Constants and conversion factors
units given in [ ]
mass of earth: ME = 5.98 x 1024 m
radius of earth: rE = 6378 km
1 mi = 1609 m
atmospheric pressure: 1 atm = 1.013 x 105 Pa
1 ft = 0.305 m
0 °C = 273.15 K
T(K) = T(C) + 273
NA = 6.022 x 1023 mol-1
T(C) = (5/9)(F -32)
R = 8.31 J/(K mol)
3
1 atm = 101.3x10 Pa = 14.7 psi
k = 1.38 x 10-23 m2 kg s-2 K-1
σ = 5.67 x 10-8 J/(sm2K4)
universal gravitation: G = 6.67 x 10-11 N m2/s2
MECHANICS: KINEMATICS, FORCES, WORK, ENERGY, POWER, MOMENTUM
displacement x [m], velocity v [m/s], acceleration a [m/s2] and time t [s]
subscript i means initial, subscript f means final
Set of equations for constant acceleration:
vf = vi + at
xf = xi + vit + 1/2at2
xf = xi + 1/2(vi + vf)t
vf2 = vi2 + 2a(xf-xi)
constant circular acceleration,
same set of equations with angular
displacement Θ, angular velocity
ω, angular acceleration α.
tangential velocity vT = s/t (s arc length, t time)
tangential acceleration aT = rα (α in rad/s2)
centripetal acceleration aC = v2/r = rω2
ω = 2π/T
(T period is time for one revolution)
vT = rω
(r radius)
Newton’s Second Law
unit for force is N [N = kgm/s2]
ΣF = ma (m mass, F force)
Στ = iα ( i moment of inertia, i = mr2 for solid sphere)
Work
[Joules J = Nm]
W = F d cos(Θ)
(d displacement, F force, Θ angle between the two)
Work = final total mechanical energy minus initial total mechanical energy
Power = Work per time (P = W/t)
unit for Power is Watt W = J/s
or Power = Force times Velocity (P = Fv)
Energy
unit for Energy is Joule J
potential gravitational energy PE = mgh (m mass, h height, g accel. due to gravity)
elastic potential energy PE = 1/2kx2
(k spring constant, x compression)
2
kinematic energy KE = 1/2mv
(m mass, v velocity)
1206 FINAL EXAM
Formula sheet - December 2009
Momentum
p = mv
Impuls J = Favg * Δt
(m mass, v velocity)
Impuls = change of momentum
Friction force:
Ff = μ FN
(μ friction coefficient and FN normal force)
Spring force:
Fs = k x
(k spring constant, x compression)
Centripetal force: FC = G mME/r2 = mv2/r
(ME mass of the earth)
Torque = magnitude of force times lever arm (τ = F*l)
Angular momentum L = i ω
[Nm]
(moment of inertia i, angular velocity ω)
FLUIDS:
Density ρ = m/V
(m mass,V volume)
Pressure P = F/A
(F applied force, A area)
pressure in dependence of depth: ΔP = ρgh
(Pascal’s principle)
Buoyant force FB = ρgV
Archimedes principle: magnitude of buoyant force = weight of displaced fluid
Equation of continuity: ρ1A1v1 = ρ2A2v2 (A area, v velocity at position 1 and 2)
Bernoulli’s equation: P1 + 1/2ρv12 + ρgh1 = P2 + 1/2ρv22 + ρgh2
SOLIDS:
linear thermal expansion of a solid: ΔL = α L0 ΔT (L length, T temperature)
Elastic deformation F/A = Y (ΔL/L0) (A cross-sectional area,Y Young’s modulus)
Shear deformation F/A = S (ΔX/L0)
Volume deformation ΔP = -B (ΔV/V0)
Stress proportional to Strain
GAS:
mparticle = Mass per mole/NA
ideal gas law
PV = nRT
Stefan-Boltzmann Law: Q/t = eσT4A
KE = 3/2 kT
vrms = sqrt(2 KE/mparticle)
(Q/t power, e emissivity, T temperature)
First law of thermodynamics ΔU = Uf - Ui = Q - W
Second law of thermodynamics ΔS = ΔQ/T
(U internal energy)
(Q heat)
1206 FINAL EXAM
Formula sheet - December 2009
Type of thermal process Work done First law of Thermodynamics (ΔU = Q - W)
Isobaric (constant pressure) W = P(Vf-Vi)
and ΔU = Q - P(Vf - Vi)
Isochoric (constant volume) W = 0 J and ΔU = Q
Isothermal (constant temperature) W = nRT ln(Vf/Vi) and 0J = Q - nRT ln(Vf/Vi)
Adiabatic (no heat flow) W = 3/2nR(Tf - Ti) and ΔU = -3/2nR(Tf - Ti)
CARNOT ENGINE:
efficiency e = 1 - QC/QH (all processes)
also e = 1 - TC/TH
(specific to Carnot process)
Work performed W = QH e
SIMPLE HARMONIC MOTION:
Formulas for simple harmonic motion (see drawing):
PENDULUM
ω = 2πf = sqrt(mgL/I)
1206 FINAL EXAM
Formula sheet - December 2009
WAVES:
velocity v = λ/T = fλ
Wave motion toward +x: y = A sin (2πft - 2πx/λ)
Wave motion toward - x: y = A sin (2πft + 2πx/λ)
Sound intensity I = P/A (Power over Area)
decibel: β = (10 dB) log (I/I0) I0 = 1.00 x 10-12 W/m2
DOPPLER EFFECT
Observer moving toward stationary source fo = fs (1 + vo/v)
Observer moving away from stationary source fo = fs (1 - vo/v)
observer and sound source are moving fo = fs { [1 ± (vo/v)]/[1 -/+ (vs/v)]}
In the numerator, the plus sign applies when the observer moves toward the source,
and the minus sign applies when the observer moves away from the source. In the
denominator, the minus sign is used when the source moves toward the observer, and
the plus sign is used when the source moves away form the observer.
Diffraction (slit): sinΘ = λ/D (circular opening sinΘ = 1.22 λ/D)
Sting fixed at both ends: fn = n(v/2L)
f2/f1 = (v2/2L)/(v1/2L) = v2/v1