Download Physics Annotated Formula Sheet

Physics Annotated Formula Sheet
Formula
Symbol and Units
d = displacement in m (meter)
Displacement
d = x – xo
x = position in m
+ or – depending on direction vav = average velocity in m/s
t = change in time in s (second)
Constant velocity
a = acceleration in m/s2
vav = d/t
v = instantaneous velocity in m/s
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
Formula
Symbol and Units
Normal force, Fn, is the  force on the object by the surface
Ff = force of friction in N
Force of friction
For static friction: Ff sFn  = coefficient of friction
For kinetic friction: Ff = kFn Fn = force normal in N
Accelerating forces problems.
Fn
Fp
Fp-
Ff
Fp-||
Fg
Fn
Fp
t
t
Graphing accelerated motion in one dimension
d
v
a
t
Vector addition
y
t
t
Fg
t
Bx= BcosB
Rx
By = BsinB
B
Ry
R
A
 Fg
Ff
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
d
vo
vt
a
t
direction
dy
vyo
vyt
-g
vertical
t
dx
vx
horizontal
vc = perimeter velocity in m/s
Uniform circular motion
r = radius of circle in m
vc = 2r/T
T = period of motion in s
ac = vc2/r
ac is directed toward center ac = centripetal acceleration in m/s2
Newton's Laws of Motion
1. Object stay in same motion unless acted upon by a force
2. Acceleration if proportional to force/mass
3. For every action there is an equal, but opposite reaction
F = force in N (Newton)
Accelerating force
F|| = ma
m = mass in kg (kilogram)
a = acceleration in m/s2
Fs = spring force in N
Spring force
Fs = kx
k = spring constant in N/m
x = distance stretched in m
Fg = force of gravity in N
Force of gravity (weight)
Fg = mg
m = mass in kg
g = 10 m/s2

Fg-||
1. Label all forces
2. resolve non-||, non- forces into || and  components
3.  F|| = ma (m is all moving mass)
4.  F = 0
m = mass of A and B in kg
Masses hanging from a
pulley, where mA > mB
g = 10 m/s2
(mA – mB)g = (mA + mB)a a = acceleration of system in m/s2
Fc = centripetal force in N
Centripetal force
m = mass in kg
Fc = mac = mv2/r
ac = centripetal acceleration in m/s2
v = perimeter velocity in m/s
r = radius of circle in m
Fg = force of gravity in N
Force of gravity between
G = 6.67 x 10-11 N•m2/kg2
planets
Fg = GMm/r2
M, m = mass in kg
r = distance between centers in m
Force of gravity is centripetal
v = perimeter velocity in m/s
GMm/r2 = mv2/r
cm = center of mass in m
Center of mass
cm = m1r1 + m2r2 + ...
m = mass in kg
(m1 + m2 + ...)
r = distance from 0 position in m
Non-accelerating force problems where forces act through cm.
1. Draw free body diagram
2. Resolve all forces into x-components and y-components
3.  Fx = 0
4.  Fy = 0
5. 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.
1.
2.
3.
4.
5.
Draw free body diagram
Determine axis of rotation that eliminates an unknown
F x r = F x r (torque)
F =  F 
 F  =  F 
Formula
Work: W = F||d
+ or – , but no direction
Power: P = W/t = Fvav
W can 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
Symbol and Units
Formula
Symbol and Units
W = work in J (Joule)
Simple harmonic motion (SHM) T = period in s
F|| = force in N
m = mass in kg
d = distance parallel to F in m
k = spring constant in N/m
P = power in W (Watt)
A = amplitude in m
K = kinetic energy in J
vo = velocity at midpoint in m/s
m = mass in kg
v = velocity in m/s
Ug = gravity potential energy in J
Time to complete one cycle
g = 10 m/s2
T = 2(m/k)½
h = height above surface in m
0
±A
displacement
G = 6.67 x 10-11 N•m2/kg2
velocity, v
vA = 0
vo = 2A/T = A(k/m)½
M = planet mass in kg
acceleration, a
ao = 0
aA = vo2/A = A(k/m)
r = distance center-center in m
2
U
=
spring
potential
energy
in
J
potential
energy,
U
U
=
0
UA = ½kA2
s
o
Us = ½kx
2
k = spring constant in N/m
kinetic energy, K
Ko = ½mvo
KA = 0
x = distance stretched in m
T = period in s
Period of a simple pendulum
Energy problems
L = length of pendulum in m
T = 2(L/g)½
1. determine initial energy of the object, Eo
g = gravity acceleration in m/s2
2. determine energy +/– due to a push or pull: Wp = ±F||d
Mechanical wave
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
p = linear momentum in kg•m/s
Linear momentum
p = mv
m = mass in kg
v = velocity in m/s
Impulse
J = impulse in N•s
J = Ft = mv = p
 amplitude, A: maximum height of a crest or depth of a trough
F = force in N
measured from the midpoint (m)
Kinetic energy to momentum
t = time in s
K = p2/2m
 wavelength, : distance between any two successive identical
K = kinetic energy in J
points of the wave (m)
Stationary  separation
 frequency, f: the number of complete waves that pass a given
0 = mAvA' + mBvB'
point per unit time (Hz or s-1)
Inelastic collision
 period, T: the time it takes for one wave to pass (s)
mAvA + mBvB = (mA + mB)v'
conservation of p, but not K
 T = 1/f
Elastic collision
 velocity, vw: speed of the waveform, vw = /T = f (m/s)
mAvA + mBvB = mAvA' + mBvB'
 transverse wave (string): disturbance  wave 
vA + vA' = vB + vB'
 longitudinal wave (sound): disturbance  wave 
conservation of p and K
Interference
Collision in two dimensions
 amplitudes combine (superposition principle)
px: mAvAx + mBvBx = (mA + mB)vx' or mAvAx' + mBvBx'
 constructive interference when amplitudes are added
py: mAvAy + mBvBy = (mA + mB)vy' or mAvAy' + mBvBy'
 destructive interference when amplitudes are subtracted
Ballistic pendulum problems
 beats, fbeats = |fA – fB|
1. bullet strikes block and sticks
vw = velocity of wave in m/s
Velocity of a wave on a string
mvm + 0 = (m + M)v'
Ft = force of tension in N
½
vw = (Ft/)
2. block swings or slides
 = linear density in kg/m
2
2
swing (K = Ug): ½(m + M)v' = (m + M)gh  h = v' /2g
Harmonics
slide (K = Wf): ½(m + M)v'2 = (m + M)gd  d = v'2/2g
Moment of Inertia (angular inertia): I = moment of inertia in kg•m2
I = mr2
m = mass in kg
r = radius of circular path in m
point mass in a circular orbit
L = angular momentum in
Angular momentum
kg•m2/s
L = I = rp = rmv
 = angular velocity in rad/s
point mass in a circular orbit
p = linear momentum in kg•m/s
Conservation of angular
v = linear velocity in m/s
momentum: r1v1 = r2v2
Determining nth harmonic
 = wavelength in m

E = energy in J
Matter energy equivalence
L = length of string in m
n = 2L/n
E = mc2
m = mass in kg
n = number of harmonic
fn = nf1
c = 3 x 108 m/s
f = frequency
Binding energy, BE
mnuclide + mBE = mp + mn
f' = perceived frequency in s-1
Doppler effect
f’
=
f(v
±
v
)/(v
±
v
)
f = generated frequency in s-1
w
o
w
s
Nuclear reactions
approaching: f' > f (+vo, –vs)
vw = wave velocity in m/s
 proton: 11p, neutron 10n, electron 0-1e, positron 01e
receding: f' < f (–vo, +vs)
vo = observer velocity in m/s
 alpha:  = 42He, beta:  = 0-1e
vs = source velocity in m/s
approximation formula
 conservation of mass # & charge: 23892U  42He + 23490Th
2

f/f

v/v
w
 nuclear process: mproducts – mreactants = mBE < 0 (E = mc )
approaching: f’ = f + f
 half life: 1  ½  ¼ take same amount of time t½
receding: f’ = f – f
Formula
Symbol and Units
Angle of reflection
i = incoming ray  to surface
i = r
r = reflected ray  to surface
phase shift when ni < nr
n = index of refraction
c = 3 x 108 m/s
Wave velocity in a vacuum
f = frequency of wave in s-1 (Hz)
c = f
 = wavelength in m
Refraction within a medium
n = index of refraction (no units)
vn = c/n
vn = velocity at n in m/s
fn = f1
n = 1/n
Angle of refraction (Snell's law) ni = source medium n
nisini = nRsinR
i = incident angle  to surface
ni < nR: bend toward normal
nR = refracting medium n
ni > nR: bend away from normal R = refracted angle  to surface
 n  to f  color separation = dispersion (prism)
 total reflection when ni > nR and i  c: sinc = nlow/nhigh
r = radius of curvature in m
Parabolic mirror radius of
curvature
r = 2f
f = focal length in m
do = object distance to l/m in m
lens/mirror equation
di = image distance to l/m in m
1/do + 1/di = 1/±f
f = focal length in m
+di for real image (-di virtual)
+f for converging (-f diverging) M = magnification (no units)
hi = height of image in m
magnification equation
ho = height of object in m
M = hi/ho = -di/do
do > +f
do < +f
–f
Formula
Density

 = m/V
Symbol and Units
 = density in kg/m3
m = mass in kg
V = volume in m3
kg/m3 = g/cm3 x 103
s.g. = specific gravity (no units)
Specific gravity
s.g. = mair/(mair – mfluid)
mair = mass measured in air
mfluid = submerged mass
object = s.g. x fluid
P = pressure in Pa (Pascals)
Pressure on a surface
P = F/A
F = force in N
A = Area in m2
Force on a hydraulic piston
PPa = Patm x 105
Fin/Ain = Fout/Aout
P = pressure in Pa
Pressure in fluid at a depth
f = density of fluid in kg/m3
P = fgh
g = 10 m/s2
h = depth in m
Upward force on a submerged Fb = buoyant force in N
object (Archimedes principle)
f = density of fluid in kg/m3
g = 10 m/s2
Fb = fgVo
Vo = object's submerged volume
V/t = volume flow rate in m3/s
Fluid flow in a pipe
A = area at a position in m2
V/t = Av = Constant
v = velocity at a position in m/s
Solve plumbing, lift & tank leak P = pressure on fluid in Pa
problems (Bernoulli's equation)  = density of fluid in kg/m3
g = 10 m/s2
P + gy + ½v2 = Constant y = elevation in m
v = velocity in m/s
Thermal expansion
L = change in length in m
 = expansion coefficient in o C-1
L = LoT
Lo = original length in m
T = temperature change in o C
K = kinetic energy in J
Kinetic energy of gases
R = 8.31 J/mol•K
3
K = /2RT
Interference with two slits
 = angle from slits to band in m
T = Temperature in K
tan = x/L
x = center to band distance in m Velocity of gas molecules
v = velocity in m/s
L = slits to screen distance in m
M = molar mass in kg
sinc = m/d
½
v = (3RT/M)
m = band order (no units)
P = pressure in Pa
sind = (m + ½)/d
V = volume in m3
Ideal gas law
c for bright band (d for dark)  = wavelength of light in m
n = number of moles
d
=
distance
between
slits
in
m
PV = nRT
Interference with one slit
TK = ToC + 273
W = width of light spot
W = 2L/d'
d' = width of slit
PV diagram
 +Win (-Wout) toward y-axis, -Win (+Wout) away from y-axis
Thickness of a film, T (f = 1/n)
 +T and +U away from origin (P x V)
ni < nf < n r
nf > ni and nr
Interference
PV (heat engine) problems
Bright
U = internal energy change in J
T = ½f
T = ¼f
U = 3/2nRT = 3/2PV = 3/2PV n = number of moles
Dark
T = ¼f
T = ½f
R = 8.31 J/mol•K
Win = -PV = Area
EM Radiation
Qin = heat added to system in J
U = Qin + Win
 High energy has short , high f (low energy has long , low f)
For complete cycle: U = 0 Win = work on the system in J
 Transverse wave polarizable
Process
T
U =
Qin
+ Win
 Doppler shift: moving away = shift to longer  (red shift)
3/ PV
0
Isometric(V = 0) PV/nR
U
E = Energy in J
Photon energy
2
3/ PV
E = hf = mc2
h = 6.63 x 10-34 J•s
Isobaric(P = 0)
PV/nR
U – Win
-PV
2
UV > violet ... red > infrared
f = frequency in s-1
0
0
-Win
-Qin
Isothermic (T = 0)
Photon momentum
m = relativistic mass in kg
Adiabatic (Q = 0)
W
0
?
U
in
p = mc = h/ = E/c
c = 3 x 108 m/s
ec = ideal efficiency (no units)
Efficiency of a heat engine
Particle wavelength (De Broglie)
 = wavelength in m
ec = (Thigh – Tlow)/Thigh
T = temperature in K
p = momentum in kg•m/s
particle = h/p
e = |Wcycle|/Qin
e = actual efficiency (no units)
Atomic energy levels (Bohr model) En = electron energy in eV
Q/t = rate of heat flow in J/s
Rate
of
heat
flow
through
a
En = -B/n2
B = 13.6 eV for hydrogen
A = area of barrier in m2
barrier
n = energy level (1, 2, etc.)
Energy absorbed by an atom
TH = high temperature in o C
Q/t  A(TH – TL)/L
EeV = photon energy in eV
EeV = En-high – En-low
TL = low temperature in o C
nm = wavelength in nm
EeV = 1240 eV•nm/nm
L = thickness of barrier
Kelectron = kinetic energy in eV Heat gain/loss by a material
Photoelectric effect
Q = heat in J
Kelectron = Ephoton - 
Ephoton = 1240 eV•nm/nm
m = mass in kg
Q = mcT
Kinetic energy of an electron
 = work function in eV
c = specific heat in J/kg•K
me = 9.11 x 10-31 kg
2
Kelectron = ½mev
v = electron velocity in m/s
Formula
Symbol and Units
Conducting sphere: excess charge on outer surface, E = 0 inside
Electric force between charges Fe = electric force in N
Fe = k|Qq|/r2
k = 9 x 109 N•m2/C2
attract for unlike (repel for like) Q, q = charge in C (Coulombs)
r = Q1 to Q2 distance in m
Electric field around a charge
E = electric field in N/C or V/m
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 field E
Fe = electric force in N
Fe = |q|E
q = charge in C
+q: E  , Fe –q: E , Fe  E = electric field in N/C
Ue = electric potential energy in J
Electric potential energy
between charges Ue = kQq/r
k = 9 x 109 N•m2/C2
+Ue for like (-Ue for unlike)
Q, q = charge in C (Coulombs)
r = Q1 to Q2 distance in m
Electric potential (voltage)
V = potential (voltage) in V (volts)
around a charge V = 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
Ue = electric potential energy in J
Electric potential energy on a
q = charge in C
charge in an electric potential
Ue = qV
V = voltage (potential) in V
Kinetic energy equals loss in Ue m = mass in kg
v = velocity in m/s
K = -Ue
½mv2 = |qV|
Current flow
I = current in A (amperes)
I = Q/t
Q = charge in C
t = time in s
Resistance in wires
R = resistance in  (ohms)
 = resistivity in •m
R = L/A
L = length in m
A = cross-section area in m2
V = terminal voltage in V
Battery terminal voltage
E = emf in V
V = E ± IR
I = current in A
+ when battery is recharging
– when battery is discharging
R = internal resistance in 
V = voltage in V
Voltage loss (Ohm's law)
V = IR
I = current in A
Power consumed
R = resistance in 
P = power in watts W
P = IV = V2/R = I2R
C = capacitance in F (farads)
Capacitor capacitance
C = єoA/d
єo = 8.85 x 10-12 C2/N•m2
A = plate area in m2
Capacitor store charge
d = plate separation in m
Q = CV
Q = charge in C
Capacitor store energy
V = voltage in V
2
2
UC = ½QV = ½CV = ½Q /C U = stored energy in joules J
C
Electric field between capacitor E = electric field in V/m
plates
E = V/d
V = voltage in V
d = distance between plates
Direction is from Vhigh  Vlow
Variable Capacitor problems
Adjust A or d
Capacitance
Battery Connection
Connected
Disconnected
C = єoA
Area Distance
(A)
(d)
d
Q = C x V
Q = C x V


















Formula
Circuit Element Symbols
Symbol and Units
Battery
Capacitor
Resistor
Summary Chart for Circuit Elements in Series and Parallel
Element
S/P
Formula
Constant Variable
Rs = R1 + R2
Series
Is
Vn = IsRn
Resistor
1/Rp = 1/R1 + 1/R2
Vp
Parallel
In = Vp/Rn
1/Cs = 1/C1 + 1/C2
Qs
Vn = Qs/Cn
Series
Capacitor
CP = C1 + C2
Vp
Qn = CnVp
Parallel
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
1. Determine overall resistance: combine Rp until all Rs
2. Determine the overall current of the circuit: I = Vtot/Rtot
3. Determine voltage loss in series resistors: V = ItotR
4. Determine voltage in parallel components: Vp = Vtot –  Vs
5. Determine I and P for each resistor: I = V/R, P = IV
6. 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)
FB = force in N
Magnetic force on a moving
charge:
FB = qvB
q = charge in C
v = velocity in m/s
Magnetic forces are centripetal
B = magnetic field in T
qvB = mv2/r
palm toward center of circle path m = mass in kg
r = radius of circular path in m
Magnetic force on current wire
I = current in A
FB = ILB
L = length of wire in m
Direction
F
B
I, v
Magnetic field near a wire
I out
I in 
B B = k'I/r B
Magnetic field in a solenoid
B out
B in 
B = magnetic field in T (teslas)
k' = 2 x 10-7 T•m/A
I = current in A
r =  distance from wire m
o = 4 x 10-7 T•m/A
N = number of turns
L = length in m
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 = flux in Wb (weber)
A = enclosed area  to B in m2
B = A x B
B = magnetic field in T
Induced emf in a wire loop
E = emf in V
B = change in flux in Wb
E = B/t
t = time in s
Induced emf in a moving rod
v = velocity of rod in m/s
L = distance between rails m
E = vLB
B = magnetic field in T
Direction of induced current
B
Induced Current

thumb
(increase: flip, decrease: no flip)
I = E/R
increase
clockwise
(rotate || to , move B closer)
Up
decrease
counter clockwise
increase
counter clockwise
Down
decrease
clockwise