Physicsannotatedformula Sheet

PhysicsAnnotatedFormula Sheet

Formula / Symbol and Units
Displacement
d = x – xo
+ or – depending on direction / d = displacement in m (meter)
x = position in m
vav= average velocity in m/s
t = change in time in s (second)
a = acceleration in m/s2
v = instantaneous velocity in m/s
Constant velocity
vav = d/t
Accelerated motion
a = (vt – vo)/t
Kinematic formulas
d = vot + ½at2
d = ½(vo + vt)t
vt = vo + at
vt2 = vo2 + 2ad
Graphing constant velocity in one dimension
d / v / a
t / t / t
Graphing accelerated motion in one dimension
d / v / a
t / t / t
Vector addition
y Rx Bx= BcosB

By = BsinB
B
Ry R
A
Ay = AsinA
Ax = AcosA
x

Ax + Bx = Rx
Ay + By = Ry
R = (Rx2 + Ry2)½
tan  = Ry/Rx = tan-1(Ry/Rx)
add 180o to  when Rx is negative

Projectile motion (g = gravitational acceleration, -10 m/s2)

vertical motion use accelerated motion formulas
horizontal motion use constant velocity formula

direction / d / vo / vt / a / t
vertical / dy / vyo / vyt / -g / t
horizontal / dx / vx
Uniform circular motion
vc = 2r/T
ac = vc2/r
ac is directed toward center / vc = perimeter velocity in m/s
r = radius of circle in m
T = period of motion in s
ac = centripetal acceleration in m/s2
Newton's Laws of Motion

Object stay in same motion unless acted upon by a force
Acceleration if proportional to force/mass
For every action there is an equal, but opposite reaction

Accelerating force
F|| = ma / F= force in N (Newton)
m = mass in kg (kilogram)
a = acceleration in m/s2
Spring force
Fs = kx / Fs = spring force in N
k = spring constant in N/m
x = distance stretched in m
Force of gravity (weight)
Fg = mg / Fg = force of gravity in N
m = mass in kg
g = 10 m/s2
Formula / Symbol and Units
Normal force, Fn, is the  force on the object by the surface
Force of friction
For static friction: FfsFn
For kinetic friction:Ff = kFn / Ff= force of friction in N
= coefficient of friction
Fn= force normal in N
Accelerating forces problems.
Fn Fp

Fp-
Ff
Fp-||

Fg
Fn
Fp

Ff  Fg
Fg

Fg-||


Label all forces
resolve non-||, non- forces into || and  components
 F|| = ma (m is all moving mass)
 F = 0

Masses hanging from a pulley, where mA > mB
(mA – mB)g = (mA + mB)a / m = mass of A and B in kg
g = 10 m/s2
a = acceleration of system in m/s2
Centripetal force
Fc = mac = mv2/r / Fc = centripetal force in N
m = mass in kg
ac = centripetal acceleration in m/s2
v = perimeter velocity in m/s
r = radius of circle in m
Force of gravity between planets
Fg = GMm/r2 / Fg= force of gravity in N
G = 6.67 x 10-11 N•m2/kg2
M, m= mass in kg
r = distance between centers in m
v = perimeter velocity in m/s
Force of gravity is centripetal
GMm/r2 = mv2/r
Center of mass
cm = m1r1 + m2r2 + ...
(m1 + m2 + ...) / cm = center of mass in m
m = mass in kg
r = distance from 0 position in m
Non-accelerating force problemswhere forces act through cm.

Draw free body diagram
Resolve all forces into x-components and y-components
 Fx = 0
 Fy = 0
3 forces, two of which are perpendicular: draw vector sum diagram and solve for missing sides of right triangle

Non-accelerating force problems where forces act away from cm.

Draw free body diagram
Determine axis of rotation that eliminates an unknown
F x r = F x r(torque)
F =  F 
F=  F 

Formula / Symbol and Units
Work:W = F||d
+ or – , but no direction / W = work in J (Joule)
F|| = force in N
d = distance parallel to F in m
P = power in W (Watt)
K = kinetic energy in J
m = mass in kg
v = velocity in m/s
Ug = gravity potential energy in J
g = 10 m/s2
h = height above surface in m
G = 6.67 x 10-11 N•m2/kg2
M = planet mass in kg
r = distance center-center in m
Us = spring potential energy in J
k = spring constant in N/m
x = distance stretched in m
Power: P = W/t = Fvav
Wcan be any energy form
Kinetic energy: K = ½mv2
Gravitational potential energy near a surface
Ug = mgh
Gravitational potential energy between planets
Ug = -GMm/r
Spring potential energy
Us = ½kx2
Energy problems
1.determine initial energy of the object, Eo
2.determine energy +/– due to a push or pull: Wp = ±F||d
3.determine energy removed by friction: Wf = Ffd
4.determine resulting energy, E'= Eo ± Wp – Wf
5.determine d, h, x or v
6.general equation: K + U ± Wp – Wf = K' + U'
½mv2 + mgh + ½kx2 ± Fpd – Ffd = ½mv'2 + mgh' + ½kx'2
Linear momentum
p = mv / p = linear momentum in kg•m/s
m = mass in kg
v = velocity in m/s
J = impulse in N•s
F = force in N
t = time in s
K = kinetic energy in J
Impulse
J = Ft = mv = p
Kinetic energy to momentum
K = p2/2m
Stationary  separation
0 = mAvA' +mBvB'
Inelastic collision
mAvA + mBvB = (mA + mB)v'
conservation of p, but not K
Elastic collision
mAvA + mBvB = mAvA' + mBvB'
vA + vA' = vB + vB'
conservation of p and K
Collision in two dimensions
px: mAvAx + mBvBx = (mA + mB)vx' or mAvAx' + mBvBx'
py: mAvAy + mBvBy = (mA + mB)vy' or mAvAy' + mBvBy'
Ballistic pendulum problems

bullet strikes block and sticks

mvm + 0 = (m + M)v'

block swings or slides

swing (K = Ug): ½(m + M)v'2 = (m + M)gh  h = v'2/2g
slide (K = Wf): ½(m + M)v'2 = (m + M)gd  d = v'2/2g
MomentofInertia(angularinertia): I = mr2
point mass in a circular orbit / I = moment of inertia in kg•m2
m = mass in kg
r = radius of circular path in m
L = angular momentum in kg•m2/s
 = angular velocity in rad/s
p = linear momentum in kg•m/s
v = linear velocity in m/s
Angular momentum
L = I =rp = rmv
point mass in a circular orbit
Conservationofangularmomentum: r1v1 = r2v2
Matter energy equivalence
E = mc2 / E = energy in J
m = mass in kg
c = 3 x 108 m/s
Binding energy, BE
mnuclide + mBE = mp + mn
Nuclear reactions

proton: 11p, neutron 10n, electron 0-1e, positron 01e
alpha: = 42He, beta: = 0-1e
conservation of mass # charge: 23892U 42He + 23490Th
nuclear process: mproducts – mreactants=mBE< 0 (E = mc2)
half life: 1  ½  ¼ take same amount of time t½

Formula / Symbol and Units
Simple harmonic motion (SHM)
Time to complete one cycle
T = 2(m/k)½ / T = period in s
m = mass in kg
k = spring constant in N/m
A = amplitude in m
vo = velocity at midpoint in m/s
displacement / 0 / ±A
velocity, v / vo = 2A/T = A(k/m)½ / vA = 0
acceleration, a / ao = 0 / aA = vo2/A = A(k/m)
potential energy, U / Uo = 0 / UA = ½kA2
kinetic energy, K / Ko = ½mvo2 / KA = 0
Period of a simple pendulum
T = 2(L/g)½ / T = period in s
L = length of pendulum in m
g = gravity acceleration in m/s2
Mechanical wave

amplitude, A: maximum height of a crest or depth of a trough measured from the midpoint (m)
wavelength, : distance between any two successive identical points of the wave (m)
frequency, f: the number of complete waves that pass a given point per unit time (Hz or s-1)
period, T: the time it takes for one wave to pass (s)
T = 1/f
velocity, vw: speed of the waveform, vw = /T = f (m/s)
transverse wave (string): disturbance  wave 
longitudinal wave (sound): disturbance  wave 

Interference

amplitudes combine (superposition principle)
constructive interference when amplitudes are added
destructive interference when amplitudes are subtracted
beats, fbeats = |fA – fB|

Velocity of a wave on a string
vw = (Ft/)½ / vw = velocity of wave in m/s
Ft = force of tension in N
 = linear density in kg/m
Harmonics

Determining nth harmonic
n = 2L/n
fn = nf1 /  = wavelength in m
L = length of string in m
n = number of harmonic
f = frequency
Doppler effect
f’ = f(vw ± vo)/(vw ± vs)
approaching: f' > f (+vo, –vs)
receding: f' < f (–vo, +vs)
approximation formula
f/f  v/vw
approaching: f’ = f + f
receding:f’ = f – f / f' = perceived frequency in s-1
f = generated frequency in s-1
vw = wave velocity in m/s
vo = observer velocityin m/s
vs= source velocity in m/s
Formula / Symbol and Units
Angle of reflection
i = r
phase shift when ni < nr / i = incoming ray  to surface
r = reflected ray  to surface
n = index of refraction
Wave velocity in a vacuum
c = f / c = 3 x 108 m/s
f = frequency of wave in s-1 (Hz)
 = wavelength in m
n = index of refraction (no units)
vn = velocity at n in m/s
Refraction within a medium
vn = c/n
fn = f1
n = 1/n
Angle of refraction (Snell's law)
nisini = nRsinR
ninR: bend toward normal
ninR: bend away from normal / ni = source medium n
i = incident angle  to surface
nR = refracting medium n
R = refracted angle  to surface

n to f color separation = dispersion (prism)
total reflection when ni > nR and ic = nlow/nhigh

Parabolic mirror radius of curvaturer = 2f / r = radius of curvature in m
f = focal length in m
lens/mirror equation
1/do + 1/di = 1/±f
+di for real image (-di virtual)
+f forconverging(-fdiverging) / do = object distance to l/m in m
di= image distance to l/m in m
f = focal length in m
M = magnification (no units)
hi = height of image in m
ho = height of object in m
magnification equation
M = hi/ho = -di/do
do > +f / do < +f / –f
Interference with two slits
tan = x/L
sinc = m/d
sind = (m + ½)/d
c for bright band (d for dark) /  = angle from slits to band in m
x = center to band distance in m
L = slits to screen distance in m
m = band order (no units)
 = wavelength of light in m
d = distance between slits in m
W = width of light spot
d' = width of slit
Interference with one slit
W = 2L/d'
Thickness of a film, T(f = 1/n)
Interference / ni < nf < nr / nf > ni and nr
Bright / T = ½f / T = ¼f
Dark / T = ¼f / T = ½f
EM Radiation

High energy has short , high f (low energy has long , low f)
Transverse wave polarizable
Doppler shift: moving away = shift to longer  (red shift)

Photon energy
E = hf = mc2
UV > violet ... red > infrared / E = Energy in J
h = 6.63 x 10-34 J•s
f = frequency in s-1
m = relativistic mass in kg
c = 3 x 108 m/s
 = wavelength in m
p = momentum in kg•m/s
Photon momentum
p = mc = h/ = E/c
Particle wavelength (De Broglie)
particle = h/p
Atomic energy levels (Bohr model)
En = -B/n2 / En= electron energy in eV
B = 13.6 eV for hydrogen
n = energy level (1, 2, etc.)
EeV = photon energy in eV
nm = wavelength in nm
Energy absorbed by an atom
EeV = En-high – En-low
EeV = 1240 eV•nm/nm
Photoelectric effect
Kelectron = Ephoton -  / Kelectron=kineticenergyineV
Ephoton = 1240 eV•nm/nm
 = work function in eV
me= 9.11 x 10-31 kg
v = electron velocity in m/s
Kinetic energy of an electron
Kelectron = ½mev2
Formula / Symbol and Units
Density
 = m/V /  = density in kg/m3
m = mass in kg
V = volume in m3
kg/m3 = g/cm3 x 103
Specific gravity
s.g. = mair/(mair – mfluid)
object = s.g. x fluid / s.g. = specific gravity (no units)
mair = mass measured in air
mfluid= submerged mass
Pressure on a surface
P = F/A / P = pressure in Pa (Pascals)
F = force in N
A = Area in m2
PPa = Patm x 105
Force on a hydraulic piston
Fin/Ain = Fout/Aout
Pressure in fluid at a depth
P = fgh / P = pressure in Pa
f = density of fluid in kg/m3
g = 10 m/s2
h = depth in m
Upward force on a submerged object (Archimedes principle)
Fb = fgVo / Fb = buoyant force in N
f = density of fluid in kg/m3
g = 10 m/s2
Vo = object'ssubmergedvolume
Fluid flow in a pipe
V/t = Av = Constant / V/t = volume flow rate in m3/s
A = area at a position in m2
v = velocity at a position in m/s
Solveplumbing,lifttankleak problems (Bernoulli's equation)
P + gy + ½v2 = Constant / P = pressure on fluid in Pa
 = density of fluid in kg/m3
g = 10 m/s2
y = elevation in m
v = velocity in m/s
Thermal expansion
L = LoT / L = change in length in m
 = expansioncoefficientinoC-1
Lo = original length in m
T = temperature change in oC
Kinetic energy of gases
K = 3/2RT / K = kinetic energy in J
R = 8.31 J/mol•K
T = Temperature in K
v = velocity in m/s
M = molar mass in kg
P = pressure in Pa
V = volume in m3
n = number of moles
TK = ToC + 273
Velocity of gas molecules
v = (3RT/M)½
Ideal gas law
PV = nRT
PV diagram

+Win (-Wout) toward y-axis, -Win(+Wout) away from y-axis
+T and +U away from origin (P x V)

PV (heat engine) problems
U=3/2nRT=3/2PV=3/2PV
Win = -PV = Area
U = Qin + Win
For complete cycle: U = 0 / U= internalenergychangeinJ
n = number of moles
R = 8.31 J/mol•K
Qin = heat added to system in J
Win= work on the system in J
Process / T / U = Qin + Win
Isometric(V = 0) / PV/nR / 3/2PV / U / 0
Isobaric(P = 0) / PV/nR / 3/2PV / U – Win / -PV
Isothermic(T=0) / 0 / 0 / -Win / -Qin
Adiabatic (Q = 0) / ? / Win / 0 / U
Efficiency of a heat engine
ec = (Thigh – Tlow)/Thigh
e = |Wcycle|/Qin / ec = ideal efficiency (no units)
T = temperature in K
e = actual efficiency (no units)
Rate of heat flow through a barrier
Q/t  A(TH – TL)/L / Q/t = rate of heat flow in J/s
A = area of barrier in m2
TH=high temperatureinoC
TL=low temperatureinoC
L = thickness of barrier
Q = heat in J
m = mass in kg
c = specific heat in J/kg•K
Heat gain/loss by a material
Q = mcT
Formula / Symbol and Units
Conducting sphere: excess charge on outer surface, E = 0 inside
Electric force between charges
Fe = k|Qq|/r2
attract for unlike (repel for like) / Fe = electric force in N
k = 9 x 109 N•m2/C2
Q, q = charge in C (Coulombs)
r = Q1 to Q2distance in m
E = electric field in N/C or V/m
Electric field around a charge
E = k|Q|/r2
away from +Q (toward -Q)
Electric field around multiple charges

Calculate E for each charge
Combine E (add for same direction, subtract for opposite direction, use Pythagorean and tan = y/x for  fields)
E = 0 between like charges and closer to lesser |Q|
E = 0 outside unlike charges and closer to lesser |Q|

Force on q in electric fieldE
Fe = |q|E
+q: E  , Fe–q: E , Fe / Fe = electric force in N
q = charge in C
E = electric field in N/C
Electric potential energy between chargesUe = kQq/r
+Ue for like (-Ue for unlike) / Ue=electricpotentialenergyinJ
k = 9 x 109 N•m2/C2
Q, q = charge in C (Coulombs)
r = Q1to Q2 distance in m
V=potential(voltage)inV(volts)
Electric potential (voltage) around a chargeV = kQ/r
+V for +Q(-V for –Q)
Electric potential around multiple charges

Calculate V for each charge
Combine V (add +V and subtract -V)
V = 0 between unlike charges and closer to lesser |Q|
V = 0 infinitely far away from like charges

Electric potential energy on a charge in an electric potential
Ue = qV / Ue=electricpotentialenergyinJ
q = charge in C
V = voltage (potential) in V
m = mass in kg
v = velocity in m/s
Kinetic energy equals loss in Ue
K = -Ue
½mv2 = |qV|
Current flow
I = Q/t / I = current in A (amperes)
Q = charge in C
t = time in s
Resistance in wires
R = L/A / R = resistance in  (ohms)
 = resistivity in •m
L = length in m
A = cross-section area in m2
Battery terminal voltage
V = E ± IR
+ when battery is recharging
– when battery is discharging / V = terminal voltage in V
E = emf in V
I = current in A
R = internal resistance in 
Voltage loss (Ohm's law)
V = IR / V = voltage in V
I = current in A
R = resistance in 
P = power in watts W
Power consumed
P = IV = V2/R = I2R
Capacitor capacitance
C = єoA/d / C = capacitance in F (farads)
єo = 8.85 x 10-12 C2/N•m2
A = plate area in m2
d = plate separation in m
Q = charge in C
V = voltage in V
UC = stored energy in joules J
Capacitor store charge
Q = CV
Capacitor store energy
UC = ½QV = ½CV2 = ½Q2/C
Electric field between capacitor platesE = V/d
Direction is from VhighVlow / E = electric field in V/m
V = voltage in V
d = distance between plates
Variable Capacitor problems
Adjust A or d / Capacitance / Battery Connection
Area
(A) / Distance
(d) / C = єoA
d / Connected / Disconnected
Q = C x V / Q = C x V
 /  /  /  /  /  / 
 /  /  /  /  /  / 
Formula / Symbol and Units
Circuit Element Symbols
Battery / Capacitor / Resistor
Summary Chart for Circuit Elements in Series and Parallel
Element / S/P / Formula / Constant / Variable
Resistor / Series / Rs = R1 + R2 / Is / Vn = IsRn
Parallel / 1/Rp = 1/R1 + 1/R2 / Vp / In = Vp/Rn
Capacitor / Series / 1/Cs = 1/C1 + 1/C2 / Qs / Vn = Qs/Cn
Parallel / CP = C1 + C2 / Vp / Qn = CnVp
Kirchhoff’s Circuit Rules

loop rule: V = 0 for any complete circuit
junction rule: Iin = Iout for any junction

General steps for solving a circuit problem

Determine overall resistance: combine Rp until all Rs
Determine the overall current of the circuit: I = Vtot/Rtot
Determine voltage loss in series resistors: V = ItotR
Determine voltage in parallel components: Vp = Vtot –  Vs
Determine I and P for each resistor: I = V/R, P = IV
Determine Q and UC for each capacitor: Q = CV, Uc = ½QV

Measuring I and V
I: place ammeter between battery and circuit element (series)
V: attach voltmeter to each side of circuit element (parallel)
Magnetic force on a moving charge: FB = qvB / FB = force in N
q = charge in C
v = velocity in m/s
B = magnetic field in T
m = mass in kg
r = radius of circular path in m
I = current in A
L = length of wire in m
Magnetic forces are centripetal
qvB = mv2/r
palmtowardcenterofcirclepath
Magnetic force on current wire
FB = ILB
Direction F B

I, v
Magnetic field near a wire
Iout Iin
B B = k'I/r B / B = magnetic field in T (teslas)
k' = 2 x 10-7T•m/A
I = current in A
r =  distance from wire m
o = 4 x 10-7T•m/A
N = number of turns
L = length in m
Magnetic field in a solenoid
B out B in
I B = oI(N/L) I
Magnetic force between wires
FB = k'I1I2L/r
Direction: I1I2 = attraction
Permanent Magnetics

Magnetic field lines go from north pole to south pole
Earth's north magnetic pole is at the south geographic pole

Magnetic flux
B = A x B / B = flux in Wb (weber)
A = enclosed area  to B in m2
B = magnetic field in T
E = emf in V
B = change in flux in Wb
t = time in s
v = velocity of rod in m/s
L = distance between rails m
B = magnetic field in T
Induced emf in a wire loop
E = B/t
Induced emf in a moving rod
E = vLB
Direction of induced current
B
thumb / 
(increase: flip, decrease: no flip) / Induced Current
I = E/R
Up / increase
(rotate || to , move B closer) / clockwise
decrease / counter clockwise
Down / increase / counter clockwise
decrease / clockwise