Physics Formula list (1)

Reference Guide & Formula Sheet for Physics
Dr. Hoselton & Mr. Price Page 1 of 8
Version 5/12/2005
#3 Components of a Vector
if V = 34 m/sec ∠48°
then
Vi
= 34 m/sec•(cos 48°); and VJ
= 34 m/sec•(sin 48°)
#4 Weight = m•g
g = 9.81m/sec² near the surface of the Earth
= 9.795 m/sec² in Fort Worth, TX
Density = mass / volume
( )3
/: mkgunit
V
m
=ρ
#7 Ave speed = distance / time = v = d/t
Ave velocity = displacement / time = v = d/t
Ave acceleration = change in velocity / time
#8 Friction Force
FF = µ•FN
If the object is not moving, you are dealing with static
friction and it can have any value from zero up to µs
FN
If the object is sliding, then you are dealing with kinetic
friction and it will be constant and equal to µK
FN
#9 Torque
τ = F•L•sin θ
Where θ is the angle between F and L; unit: Nm
#11 Newton's Second Law
Fnet = ΣFExt = m•a
#12 Work = F•D•cos θ
Where D is the distance moved and
θ is the angle between F and the
direction of motion,
unit : J
#16 Power = rate of work done
unit : watt
Efficiency = Workout / Energyin
Mechanical Advantage = force out / force in
M.A. = Fout / Fin
#19 Constant-Acceleration Linear Motion
v = vο + a•t x
(x-xο) = vο•t + ½•a•t² v
v ² = vο² + 2•a• (x - xο) t
(x-xο) = ½•( vο + v) •t a
(x-xο) = v•t - ½•a•t² vο
#20 Heating a Solid, Liquid or Gas
Q = m•c•∆T (no phase changes!)
Q = the heat added
c = specific heat.
∆T = temperature change, K
#21 Linear Momentum
momentum = p = m•v = mass • velocity
momentum is conserved in collisions
#23 Center of Mass – point masses on a line
xcm = Σ(mx) / Mtotal
#25 Angular Speed vs. Linear Speed
Linear speed = v = r•ω = r • angular speed
#26 Pressure under Water
P = ρ•g•h
h = depth of water
ρ = density of water
#28 Universal Gravitation
2
21
r
mm
GF =
G = 6.67 E-11 N m² / kg²
#29 Mechanical Energy
PEGrav = P = m•g•h
KELinear = K = ½•m•v²
#30 Impulse = Change in Momentum
F•∆t = ∆(m•v)
#31 Snell's Law
n1•sin θ1 = n2•sin θ2
Index of Refraction
n = c / v
c = speed of light = 3 E+8 m/s
#32 Ideal Gas Law
P•V = n•R•T
n = # of moles of gas
R = gas law constant
= 8.31 J / K mole.
#34 Periodic Waves
v = f •λ
f = 1 / T T = period of wave
#35 Constant-Acceleration Circular Motion
ω = ωο + α•t θ
θ−θο= ωο•t + ½•α•t² ω
ω
2
= ωο
2
+ 2•α•(θ−θο) t
θ−θο = ½•(ωο + ω)•t α
θ−θο = ω•t - ½•α•t² ωο
time
Work
Power =

Version 5/12/2005
#36 Buoyant Force - Buoyancy
FB = ρ•V•g = mDisplaced fluid•g = weightDisplaced fluid
ρ = density of the fluid
V = volume of fluid displaced
#37 Ohm's Law
V = I•R
V = voltage applied
I = current
R = resistance
Resistance of a Wire
R = ρ•L / Ax
ρ = resistivity of wire material
L = length of the wire
Ax = cross-sectional area of the wire
#39 Heat of a Phase Change
Q = m•L
L = Latent Heat of phase change
#41 Hooke's Law
F = k•x
Potential Energy of a spring
W = ½•k•x² = Work done on spring
#42 Electric Power
P = I²•R = V ² / R = I•V
#44 Speed of a Wave on a String
L
mv
T
2
=
T = tension in string
m = mass of string
L = length of string
#45 Projectile Motion
Horizontal: x-xο= vο•t + 0
Vertical: y-yο = vο•t + ½•a•t²
#46 Centripetal Force
rm
r
mv
F 2
2
ω==
#47 Kirchhoff’s Laws
Loop Rule: ΣAround any loop ∆Vi = 0
Node Rule: Σat any node Ii = 0
#51 Minimum Speed at the top of a
Vertical Circular Loop
rgv =
#53 Resistor Combinations
SERIES
Req = R1 + R2+ R3+. . .
PARALLEL
∑=
=+++=
n
i ineq RRRRR 121
11111
K
#54 Newton's Second Law and
Rotational Inertia
τ = torque = I•α
I = moment of inertia = m•r² (for a point mass)
(See table in Lesson 58 for I of 3D shapes.)
#55 Circular Unbanked Tracks
mg
r
mv
µ=
2
#56 Continuity of Fluid Flow
Ain•vin = Aout•vout A= Area
v = velocity
#58 Moment of Inertia - I
cylindrical hoop m•r2
solid cylinder or disk ½ m•r2
solid sphere 2
/5 m•r2
hollow sphere ⅔ m•r2
thin rod (center) 1
/12 m•L2
thin rod (end) ⅓ m•L2
#59 Capacitors Q = C•V
Q = charge on the capacitor
C = capacitance of the capacitor
V = voltage applied to the capacitor
RC Circuits (Discharging)
Vc = Vo•e
− t/RC
Vc − I•R = 0
#60 Thermal Expansion
Linear: ∆L = Lo
•α•∆T
Volume: ∆V = Vo
•β•∆T
#61 Bernoulli's Equation
P + ρ•g•h + ½•ρ•v ² = constant
QVolume Flow Rate = A1•v1 = A2•v2 = constant
#62 Rotational Kinetic Energy (See LEM, pg 8)
KErotational = ½•I•ω
2
= ½•I• (v / r)2
KErolling w/o slipping = ½•m•v2
+ ½•I•ω
2
Angular Momentum = L = I•ω = m•v•r•sin θ
Angular Impulse equals
CHANGE IN Angular Momentum
∆L = τorque•∆t = ∆(I•ω)

Version 5/12/2005
#63 Period of Simple Harmonic Motion
k
m
T π2= where k = spring constant
f = 1 / T = 1 / period
#64 Banked Circular Tracks
v2
= r•g•tan θ
#66 First Law of Thermodynamics
∆U = QNet + WNet
Change in Internal Energy of a system =
+Net Heat added to the system
+Net Work done on the system
Flow of Heat through a Solid
∆Q / ∆t = k•A•∆T / L
k = thermal conductivity
A = area of solid
L = thickness of solid
#68 Potential Energy stored in a Capacitor
P = ½•C•V²
RC Circuit formula (Charging)
Vc
= Vcell•(1 − e
− t / RC
)
R•C = τ = time constant
Vcell
- Vcapacitor − I•R = 0
#71 Simple Pendulum
g
L
T π2= and f = 1/ T
#72 Sinusoidal motion
x = A•cos(ω•t) = A•cos(2•π•f •t)
ω = angular frequency
f = frequency
#73 Doppler Effect
s
Toward
Away
o
Toward
Away
v
v
ff
m343
343 ±
=′
vo
= velocity of observer: vs
= velocity of source
#74 2nd
Law of Thermodynamics
The change in internal energy of a system is
∆U = QAdded + WDone On – Qlost – WDone By
Maximum Efficiency of a Heat Engine
(Carnot Cycle) (Temperatures in Kelvin)
#75 Thin Lens Equation
f = focal length
i = image distance
o = object distance
Magnification
M = −Di / Do = −i / o = Hi / Ho
Helpful reminders for mirrors and lenses
Focal Length of: positive negative
mirror concave convex
lens converging diverging
Object distance = o all objects
Object height = Ho all objects
Image distance = i real virtual
Image height = Hi virtual, upright real, inverted
Magnification virtual, upright real, inverted
#76 Coulomb's Law
2
21
r
qq
kF =
2
2
99
4
1
C
mN
Ek
o
⋅
==
πε
#77 Capacitor Combinations
PARALLEL
Ceq = C1 + C2+ C3 + …
SERIES
∑=
=+++=
n
i ineq CCCCC 121
11111
K
#78 Work done on a gas or by a gas
W = P•∆V
#80 Electric Field around a point charge
2
r
q
kE =
2
2
99
4
1
C
mN
Ek
o
⋅
==
πε
#82 Magnetic Field around a wire
r
I
B o
π
µ
2
=
Magnetic Flux
Φ = B•A•cos θ
Force caused by a magnetic field
on a moving charge
F = q•v•B•sin θ
#83 Entropy change at constant T
∆S = Q / T
(Phase changes only: melting, boiling, freezing, etc)
%100)1(% ⋅−=
h
c
T
T
Eff
ioDDf io
11111
+=+=

Version 5/12/2005
#84 Capacitance of a Capacitor
C = κ•εo•A / d
κ = dielectric constant
A = area of plates
d = distance between plates
εo
= 8.85 E(-12) F/m
#85 Induced Voltage N = # of loops
t
NEmf
∆
∆Φ
=
Lenz’s Law – induced current flows to create a B-field
opposing the change in magnetic flux.
#86 Inductors during an increase in current
VL
= Vcell
•e
− t / (L / R)
I = (Vcell
/R)•[ 1 - e
− t / (L / R)
]
L / R = τ = time constant
#88 Transformers
N1
/ N2
= V1
/ V2
I1
•V1
= I2•V2
#89 Decibel Scale
B (Decibel level of sound) = 10 log ( I / Io
)
I = intensity of sound
Io
= intensity of softest audible sound
#92 Poiseuille's Law
∆P = 8•η•L•Q/(π•r
4
)
η = coefficient of viscosity
L = length of pipe
r = radius of pipe
Q = flow rate of fluid
Stress and Strain
Y or S or B = stress / strain
stress = F/A
Three kinds of strain: unit-less ratios
I. Linear: strain = ∆L / L
II. Shear: strain = ∆x / L
III. Volume: strain = ∆V / V
#93 Postulates of Special Relativity
1. Absolute, uniform motion cannot be
detected.
2. No energy or mass transfer can occur
at speeds faster than the speed of light.
#94 Lorentz Transformation Factor
2
2
1
c
v
−=β
#95 Relativistic Time Dilation
∆t = ∆to
/ β
#96 Relativistic Length Contraction
∆x = β•∆xo
Relativistic Mass Increase
m = mo
/ β
#97 Energy of a Photon or a Particle
E = h•f = m•c2
h = Planck's constant = 6.63 E(-34) J sec
f = frequency of the photon
#98 Radioactive Decay Rate Law
A = Ao•e
− k t
= (1/2n
)•A0 (after n half-lives)
Where k = (ln 2) / half-life
#99 Blackbody Radiation and
the Photoelectric Effect
E= n•h•f where h = Planck's constant
#100 Early Quantum Physics
Rutherford-Bohr Hydrogen-like Atoms
or
R = Rydberg's Constant
= 1.097373143 E7 m-1
ns = series integer (2 = Balmer)
n = an integer > ns
Mass-Energy Equivalence
mv = mo / β
Total Energy = KE + moc2
= moc2
/ β
Usually written simply as E = m c2
de Broglie Matter Waves
For light: Ep = h•f = h•c / λ = p•c
Therefore, momentum: p = h / λ
Similarly for particles, p = m•v = h / λ,
so the matter wave's wavelength must be
λ = h / m v
Energy Released by Nuclear
Fission or Fusion Reaction
E = ∆mo•c2
Hz
nn
cR
c
f
s








−== 22
11
λ
1
22
111 −








−⋅= meters
nn
R
sλ

Version 5/12/2005
MISCELLANEOUS FORMULAS
Quadratic Formula
if a x²+ b x + c = 0
then
a
acbb
x
2
42
−±−
=
Trigonometric Definitions
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
sec θ = 1 / cos θ = hyp / adj
csc θ = 1 / sin θ = hyp / opp
cot θ = 1 / tan θ = adj / opp
Inverse Trigonometric Definitions
θ = sin-1
(opp / hyp)
θ = cos-1
(adj / hyp)
θ = tan-1
(opp / adj)
Law of Sines
a / sin A = b / sin B = c / sin C
or
sin A / a = sin B / b = sin C / c
Law of Cosines
a2
= b2
+ c2
- 2 b c cos A
b2
= c2
+ a2
- 2 c a cos B
c² = a² + b² - 2 a b cos C
T-Pots
For the functional form
CBA
111
+=
You may use "The Product over the Sum" rule.
CB
CB
A
+
⋅
=
For the Alternate Functional form
CBA
111
−=
You may substitute T-Pot-d
CB
CB
BC
CB
A
−
⋅
−=
−
⋅
=
Fundamental SI Units
Unit Base Unit Symbol
…………………….
Length meter m
Mass kilogram kg
Time second s
Electric
Current ampere A
Thermodynamic
Temperature kelvin K
Luminous
Intensity candela cd
Quantity of
Substance moles mol
Plane Angle radian rad
Solid Angle steradian sr or str
Some Derived SI Units
Symbol/Unit Quantity Base Units
…………………….
C coulomb Electric Charge A•s
F farad Capacitance A2
•s4/(kg•m2
)
H henry Inductance kg•m2
/(A2
•s2
)
Hz hertz Frequency s-1
J joule Energy & Work kg•m2
/s2
= N•m
N newton Force kg•m/s2
Ω ohm Elec Resistance kg•m2
/(A2
•s2
)
Pa pascal Pressure kg/(m•s2
)
T tesla Magnetic Field kg/(A•s2
)
V volt Elec Potential kg•m2
/(A•s3
)
W watt Power kg•m2
/s3
Non-SI Units
o
C degrees Celsius Temperature
eV electron-volt Energy, Work

Version 5/12/2005
Aa acceleration, Area, Ax=Cross-sectional Area,
Amperes, Amplitude of a Wave, Angle,
Bb Magnetic Field, Decibel Level of Sound,
Angle,
Cc specific heat, speed of light, Capacitance,
Angle, Coulombs, o
Celsius, Celsius
Degrees, candela,
Dd displacement, differential change in a variable,
Distance, Distance Moved, distance,
Ee base of the natural logarithms, charge on the
electron, Energy,
Ff Force, frequency of a wave or periodic motion,
Farads,
Gg Universal Gravitational Constant, acceleration
due to gravity, Gauss, grams, Giga-,
Hh depth of a fluid, height, vertical distance,
Henrys, Hz=Hertz,
Ii Current, Moment of Inertia, image distance,
Intensity of Sound,
Jj Joules,
Kk K or KE = Kinetic Energy, force constant of
a spring, thermal conductivity, coulomb's
law constant, kg=kilograms, Kelvins,
kilo-, rate constant for Radioactive
decay =1/τ=ln2 / half-life,
Ll Length, Length of a wire, Latent Heat of
Fusion or Vaporization, Angular
Momentum, Thickness, Inductance,
Mm mass, Total Mass, meters, milli-, Mega-,
mo=rest mass, mol=moles,
Nn index of refraction, moles of a gas, Newtons,
Number of Loops, nano-,
Oo
Pp Power, Pressure of a Gas or Fluid, Potential
Energy, momentum, Power, Pa=Pascal,
Qq Heat gained or lost, Maximum Charge on a
Capacitor, object distance, Flow Rate,
Rr radius, Ideal Gas Law Constant, Resistance,
magnitude or length of a vector,
rad=radians
Ss speed, seconds, Entropy, length along an arc,
Tt time, Temperature, Period of a Wave, Tension,
Teslas, t1/2=half-life,
Uu Potential Energy, Internal Energy,
Vv velocity, Velocity, Volume of a Gas, velocity of
wave, Volume of Fluid Displaced, Voltage, Volts,
Ww weight, Work, Watts, Wb=Weber,
Xx distance, horizontal distance, x-coordinate
east-and-west coordinate,
Yy vertical distance, y-coordinate,
north-and-south coordinate,
Zz z-coordinate, up-and-down coordinate,
Αα Alpha angular acceleration, coefficient of
linear expansion,
Ββ Beta coefficient of volume expansion,
Lorentz transformation factor,
Χχ Chi
∆δ Delta ∆=change in a variable,
Εε Epsilon εο = permittivity of free space,
Φφ Phi Magnetic Flux, angle,
Γγ Gamma surface tension = F / L,
1 / γ = Lorentz transformation factor,
Ηη Eta
Ιι Iota
ϑϕ Theta and Phi lower case alternates.
Κκ Kappa dielectric constant,
Λλ Lambda wavelength of a wave, rate constant
for Radioactive decay =1/τ=ln2/half-life,
Μµ Mu friction, µo = permeability of free space,
micro-,
Νν Nu alternate symbol for frequency,
Οο Omicron
Ππ Pi 3.1425926536…,
Θθ Theta angle between two vectors,
Ρρ Rho density of a solid or liquid, resistivity,
Σσ Sigma Summation, standard deviation,
Ττ Tau torque, time constant for a exponential
processes; eg τ=RC or τ=L/R or τ=1/k=1/λ,
Υυ Upsilon
ςϖ Zeta and Omega lower case alternates
Ωω Omega angular speed or angular velocity,
Ohms
Ξξ Xi
Ψψ Psi
Ζζ Zeta

Version 5/12/2005
Values of Trigonometric Functions
for 1st
Quadrant Angles
(simple mostly-rational approximations)
θ sin θ cos θ tan θ
0o
0 1 0
10o
1/6 65/66 11/65
15o
1/4 28/29 29/108
20o
1/3 16/17 17/47
29o
151/2
/8 7/8 151/2
/7
30o
1/2 31/2
/2 1/31/2
37o
3/5 4/5 3/4
42o
2/3 3/4 8/9
45o
21/2
/2 21/2
/2 1
49o
3/4 2/3 9/8
53o
4/5 3/5 4/3
60 31/2
/2 1/2 31/2
61o
7/8 151/2
/8 7/151/2
70o
16/17 1/3 47/17
75o
28/29 1/4 108/29
80o
65/66 1/6 65/11
90o
1 0 ∞
(Memorize the Bold rows for future reference.)
Derivatives of Polynomials
For polynomials, with individual terms of the form Axn
,
we define the derivative of each term as
To find the derivative of the polynomial, simply add the
derivatives for the individual terms:
Integrals of Polynomials
For polynomials, with individual terms of the form Axn
,
we define the indefinite integral of each term as
To find the indefinite
integral of the polynomial, simply add the integrals for
the individual terms and the constant of integration, C.
Prefixes
Factor Prefix Symbol Example
1018
exa- E 38 Es (Age of
the Universe
in Seconds)
1015
peta- P
1012
tera- T 0.3 TW (Peak
power of a
1 ps pulse
from a typical
Nd-glass laser)
109
giga- G 22 G$ (Size of
Bill & Melissa
Gates’ Trust)
106
mega- M 6.37 Mm (The
radius of the
Earth)
103
kilo- k 1 kg (SI unit
of mass)
10-1
deci- d 10 cm
10-2
centi- c 2.54 cm (=1 in)
10-3
milli- m 1 mm (The
smallest
division on a
meter stick)
10-6
micro- µ
10-9
nano- n 510 nm (Wave-
length of green
light)
10-12
pico- p 1 pg (Typical
mass of a DNA
sample used in
genome
studies)
10-15
femto- f
10-18
atto- a 600 as (Time
duration of the
shortest laser
pulses)
( ) 1−
= nn
nAxAx
dx
d
( ) 66363 2
+=−+ xxx
dx
d
( ) 1
1
1 +
+
=∫
nn
Ax
n
dxAx
( ) [ ]∫ ++=+ Cxxdxx 6366 2

Version 5/12/2005
Linear Equivalent Mass
Rotating systems can be handled using the linear forms
of the equations of motion. To do so, however, you must
use a mass equivalent to the mass of a non-rotating
object. We call this the Linear Equivalent Mass (LEM).
(See Example I)
For objects that are both rotating and moving linearly,
you must include them twice; once as a linearly moving
object (using m) and once more as a rotating object
(using LEM). (See Example II)
The LEM of a rotating mass is easily defined in terms of
its moment of inertia, I.
LEM = I/r2
For example, using a standard table of Moments of
Inertia, we can calculate the LEM of simple objects
rotating on axes through their centers of mass:
I LEM
Cylindrical hoop mr2
m
Solid disk ½mr2
½m
Hollow sphere 2
⁄5mr2 2
⁄5m
Solid sphere ⅔mr2
⅔m
Example I
A flywheel, M = 4.80 kg and r = 0.44 m, is wrapped
with a string. A hanging mass, m, is attached to the end
of the string.
When the
hanging mass is
released, it
accelerates
downward at
1.00 m/s2
. Find
the hanging
mass.
To handle this problem using the linear form of
Newton’s Second Law of Motion, all we have to do is
use the LEM of the flywheel. We will assume, here, that
it can be treated as a uniform solid disk.
The only external force on this system is the weight of
the hanging mass. The mass of the system consists of
the hanging mass plus the linear equivalent mass of the
fly-wheel. From Newton’s 2nd
Law we have
F = ma, therefore, mg = [m + (LEM=½M)]a
mg = [m + ½M] a
(mg – ma) = ½M a
m(g − a) = ½Ma
m = ½•M•a / (g − a)
m = ½• 4.8 • 1.00 / (9.81 − 1)
m = 0.27 kg
If a = g/2 = 4.905 m/s2
, m = 2.4 kg
If a = ¾g = 7.3575 m/s2
, m = 7.2 kg
Note, too, that we do not need to know the radius unless
the angular acceleration of the fly-wheel is requested. If
you need α, and you have r, then α = a/r.
Example II
Find the kinetic energy of a disk, m = 6.7 kg, that is
moving at 3.2 m/s while rolling without slipping along a
flat, horizontal surface. (IDISK = ½mr2
; LEM = ½m)
The total kinetic energy consists of the linear kinetic
energy, KL = ½mv2
, plus the rotational kinetic energy,
KR = ½(I)(ω)2
= ½(I)(v/r)2
= ½(I/r2
)v2
= ½(LEM)v2
.
KE = ½mv2
+ ½•(LEM=½m)•v2
KE = ½•6.7•3.22
+ ½•(½•6.7)•3.22
KE = 34.304 + 17.152 = 51 J
Final Note:
This method of incorporating rotating objects into the
linear equations of motion works in every situation I’ve
tried; even very complex problems. Work your problem
the classic way and this way to compare the two. Once
you’ve verified that the LEM method works for a
particular type of problem, you can confidently use it for
solving any other problem of the same type.

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