Physics Annotated Formula Sheet
Formula / Symbol and UnitsDisplacement
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 = vot + 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= BcosqB
By = BsinqB
B
Ry R
A
Ay = AsinqA
Ax = AcosqA
x
· Ax + Bx = Rx
· Ay + By = Ry
· R = (Rx2 + Ry2)½
· tan q = Ry/Rx \ q = tan-1(Ry/Rx)
· add 180o to q 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 = 2pr/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
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
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 £ msFn
For kinetic friction: Ff = mkFn / Ff = force of friction in N
m = coefficient of friction
Fn = force normal in N
Accelerating forces problems.
Fn Fp
Fp-^
Ff
Fp-||
Fg
Fn
Fp
Ff q Fg
Fg
Fg-||
q
1. Label all forces
2. resolve non-||, non-^ forces into || and ^ components
3. S F|| = ma (m is all moving mass)
4. S 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 problems where forces act through cm.
1. Draw free body diagram
2. Resolve all forces into x-components and y-components
3. S Fx = 0
4. S 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. Draw free body diagram
2. Determine axis of rotation that eliminates an unknown
3. SPF x r = SQF x r (torque)
4. S F® = S F ¬
5. S F = S 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
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
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 = FDt = mDv = Dp
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
1. bullet strikes block and sticks
mvm + 0 = (m + M)v'
2. 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 + M)gd \ d = v'2/2mg
Moment of Inertia (angular inertia): 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
v = angular velocity in rad/s
p = linear momentum in kg•m/s
v = linear velocity in m/s
Angular momentum
L = Iv = rp = rmv
point mass in a circular orbit
Conservation of angular momentum: 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: a = 42He, beta: b = 0-1e
· conservation of mass # charge: 23892U ® 42He + 23490Th
· nuclear process: mproducts – mreactants = mBE < 0 (E = Dmc2)
· 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 = 2p(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 = 2pA/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 = 2p(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, l: 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 = l/T = lf (m/s)
· transverse wave (string): disturbance E wave D
· longitudinal wave (sound): disturbance D wave D
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/a)½ / vw = velocity of wave in m/s
Ft = force of tension in N
a = linear density in kg/m
Harmonics
Determining nth harmonic
ln = 2L/n
fn = nf1 / l = 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
Df/f » v/vw
approaching: f’ = f + Df
receding: f’ = f – Df / f' = perceived frequency in s-1
f = generated frequency in s-1
vw = wave velocity in m/s
vo = observer velocity in m/s
vs = source velocity in m/s
Formula / Symbol and Units
Angle of reflection
qi = qr
phase shift when ni < nr / qi = incoming ray ^ to surface
qr = reflected ray ^ to surface
n = index of refraction
Wave velocity in a vacuum
c = fl / c = 3 x 108 m/s
f = frequency of wave in s-1 (Hz)
l = 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
ln = l/n
Angle of refraction (Snell's law)
nisinqi = nRsinqR
ni < nR: bend toward normal
ni > nR: bend away from normal / ni = source medium n
qi = incident angle ^ to surface
nR = refracting medium n
qR = refracted angle ^ to surface
· n µ to f \ color separation = dispersion (prism)
· total reflection when ni > nR and qi ³ qc = nlow/nhigh
Parabolic mirror radius of curvature r = 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 for converging (-f diverging) / 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
tanq = x/L
sinqc = ml/d
sinqd = (m + ½)l/d
qc for bright band (qd for dark) / q = 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)
l = 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 = 2lL/d'
Thickness of a film, T (lf = l1/n)
Interference / ni < nf < nr / nf > ni and nr
Bright / T = ½lf / T = ¼lf
Dark / T = ¼lf / T = ½lf
EM Radiation
· High energy has short l, high f (low energy has long l, low f)
· Transverse wave \polarizable
· Doppler shift: moving away = shift to longer l (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
l = wavelength in m
p = momentum in kg•m/s
Photon momentum
p = mc = h/l = E/c
Particle wavelength (De Broglie)
lparticle = 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
lnm = wavelength in nm
Energy absorbed by an atom
EeV = En-high – En-low
EeV = 1240 eV•nm/lnm
Photoelectric effect
Kelectron = Ephoton - f / Kelectron = kinetic energy in eV
Ephoton = 1240 eV•nm/lnm
f = 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
r = m/V / r = density in kg/m3
m = mass in kg
V = volume in m3
rkg/m3 = rg/cm3 x 103
Specific gravity
s.g. = mair/(mair – mfluid)
robject = s.g. x rfluid / 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 = rfgh / P = pressure in Pa
rf = density of fluid in kg/m3
g = 10 m/s2
h = depth in m
Upward force on a submerged object (Archimedes principle)