Equations Flashcards
dot product
A . B = |A| |B| cos θ
cross product
A x B = |A| |B| sin θ
velocity (v)
vector,
v = ∆X/∆t (m/s)
gravitational force between two objects
Fg = (Gm1m2)/r^2
where:
G = 6.67E-11 N*m^2/kg^2
static friction (fs)
fs ≤ μs*N
μs = coefficient of static friction (depends on two materials)
N = normal force, component force perpendicular to plane of contact
kinetic friction (fk)
fk = μk*N μk = coefficient of kinetic friction (depends on two materials)
weight (W)
W = m*g
m = mass
g (Fg) = 9.8 m/s^2 (approximately 10) (on earth)
center of mass/gravity of uniform object
x = ( (m1x1) + (m2x2) + (m3x3) + …) / (m1 + m2 + m3 + …)
same for y and z, just replace x
Newton’s laws
F = m*a , Fab = -Fba F = force m = mass a = acceleration Fab = force from a to b -Fba = equal and opposite reaction
acceleration (a)
vector,
a = ∆v/∆t (m/s^2)
equations of linear motion
x = v*t x = v(o)*t + (1/2) a*t^2 v = v(o) + a*t v^2 = v(o)^2 + 2*a*x
where v(o) = velocity initial
centripetal acceleration (Fc)
Fc = m*v^2 / r
torque (𝜏)
𝜏 = r*F = r*F sin θ r = length of lever arm F = magnitude of force θ = angle between lever arm and force vectors
kinetic energy (K)
K = 1/2 mv^2
unit: J = kg*m^2/s^2
gravitational potential energy (U)
U = mgh
unit: J = kg*m^2/s^2
elastic potential energy (U)
U = 1/2 kx^2
unit: J = kg*m^2/s^2
total mechanical energy (E)
E = U + K
conservation of mechanical energy
∆E = ∆U + ∆K = 0 W(conservative) = 0
work (W)
W = Fd cos θ
unit: J = kg*m^2/s^2
work for isobaric (constant pressure) process
W = P∆V
power (P)
P = W/t = ∆E/t
unit: watt (W) = J/s = kg*m^2/s^3
work-energy theorem
W(net) = ∆K = K(f) - K(i)
mechanical advantage
mechanical advantage = F(out) / F(in)
pulleys- what is relationship between tension and weight
T(total) = W
where T = tension
and W = mg
efficiency
efficiency = W(out) / W(in) = (load)(load distance) / (effort)(effort distance)
thermal expansion of solid
∆L = αL∆T
where:
L = initial length
α = coefficient of linear expansion (K^-1) = 1/3 β
thermal expansion of liquid
∆V = βV∆T
where:
V = initial volume
β = coefficient of volumetric expansion (K^-1) = 3 α
change in internal energy (U)
∆U = q - w
specific heat (c)
q = mc∆T
units of c = 1 cal/gK = 4.184 J/gK
heat of transformation/latent heat (L)
q = mL
entropy (S)
∆S = q(rev) / T
unit of S = J/mol*K
density (ρ)
ρ = m/V
pressure
P = F/A
absolute pressure
P = P(o) + ρgz
where: P(o) = incident/ambient pressure (at surface) ρ = density g = acceleration due to gravity z = depth of object
gauge pressure
P(gauge) = P - P(atm) = (P(o) + ρgz) - P(atm)
where: P(o) = incident/ambient pressure (at surface) ρ = density g = acceleration due to gravity z = depth of object below some point
Poiseuille’s law
Q = (π (r^4) ΔP) / (8ηL)
where: Q = flow rate r = radius of tube ΔP = pressure gradient η = viscosity L = length of pipe
critical speed (V(c))
V(c) = (N(R) η) / (ρD)
where: N(R) = Reynold's number η = viscosity ρ = density D = diameter of tube
continuity equation
Q = v(1) * A(1) = v(2) * A(2)
where:
Q = flow rate
v = linear speed
A = cross-sectional area
Bernoulli’s equation
P(1) + 1/2 ρv(1)^2 + ρgh(1) = P(2) + 1/2 ρv(2)^2 + ρgh(2)
where: P = absolute pressure ρ = density v = linear speed g = acceleration due to gravity h = height of fluid above some point
dynamic pressure
1/2 ρv^2
static pressure
P + ρgh
Coulomb’s law
F(e) = (k q1 q2) / r^2
where:
k = Coulomb’s constant (8.99 x 10^9 N*m^2/C^2)
q1 and q2 = magnitude of charges
r = distance between charges
electric field (E)
E = F(e) / q = kQ / r^2
where:
q = test charge
Q = source charge
F(e) = magnitude of force felt by point charge
k = Coulomb’s constant (8.99 x 10^9 N*m^2/C^2)
r = distance between charges
electric potential energy (U)
U = kQq / r
where: k = Coulomb's constant (8.99 x 10^9 N*m^2/C^2) Q = source charge q = test charge r = distance between charges
electric potential (V)
V = U / q = kQ / r
where:
k = Coulomb’s constant (8.99 x 10^9 N*m^2/C^2)
Q = source charge
r = distance between charges
potential difference (voltage)
∆V = V(b) - V(a) = W(ab) / q
where:
W(ab) = work needed to move a test charge q through an electric field from point a to b
dipole moment (p)
p = qd
where:
q = test charge
d = separation distance
net torque on a dipole (𝜏)
𝜏 = pE sin θ
where:
p = magnitude of dipole moment
E = magnitude of uniform external electric field
θ = angle the dipole moment makes with the electric field
magnetic field (B) at distance r from a wire
B = μ(o)*I / 2πr
where:
μ(o) = permeability of free space
I = current through wire
magnetic force (F(B))
F(B) = qvB sin θ
where:
v = magnitude of velocity
B = magnitude of magnetic field
θ = smallest angle between velocity and magnetic field vectors (v and B)
for a straight wire, magnitude of force created by external magnetic field (F(B))
F(B) = ILB sin θ
where:
I = current
L = length of wire in field
θ = angle between L and B
magnitude of current (I)
I = Q / ∆t
where:
Q = charge passing through conductor
junction rule
I (into junction) = I (out of junction)
loop rule
V (source) = V (drop)
resistance of a resistor (R)
R = ρL / A
where:
ρ = resistivity
L = length
A = cross-sectional area
Ohm’s law
V = IR
where:
V = voltage drop
I = current
R = resistance
power (P)
P = W / t = ∆E / t
P = IV = I^2 R = V^2 / R
total voltage/resistance of resistors in series
V(s) = V(1) + V(2) + V(3) + … + V(n)
R(s) = R(1) + R(2) + R(3) + … + R(n)
total voltage/resistance of resistors in parallel
V(p) = V(1) = V(2) = V(3) = … = V(n)
1/R(p) = 1/R(1) = 1/R(2) = 1/R(3) = … = 1/R(n)
capacitance (C)
C = Q / V
where:
Q = charge stores on one plate
V = potential difference (voltage) across capacitor
C = ε(o) A / d where: ε(o) = permittivity of free space (8.85 x 10^-12 F/m) A = area of overlap between plates d = distance between plates
potential energy stored in a capacitor
U = 1/2 CV^2
where:
C = capacitance
V = potential difference (voltage)
capacitors in series
1/C(s) = 1/C(1) + 1/C(2) + 1/C(3) + … + 1/C(n)
capacitors in parallel
C(p) = C(1) + C(2) + C(3) + … + C(n)
capacitance with dielectric material
C’ = 𝜅C
where:
C’ = new capacitance with dielectric
𝜅 = dielectric constant
C = original capacitance
frequency (ƒ)
v = ƒλ
where:
v = propagation speed
λ = wavelength
period (T)
T = 1/ƒ
where:
ƒ = frequency
angular frequency (ω)
ω = 2πƒ = 2π/T
where:
ƒ = frequency
T = period
speed of sound through a medium (v)
v = √(B/ρ)
where:
B = Bulk modulus
ρ = density
Doppler effect
ƒ’ = ƒ ((v ± v(D)) / (v ∓ v(S))
where: ƒ' = perceived frequency ƒ = actual frequency v = speed of sound in medium v(D) = speed of detector v(S) = speed of source
signs:
top sign- used when source and detector moving toward one another
bottom sign- used when source and detector moving away from one another
equation that relates wavelength of standing wave and length of string or open pipe:
λ = 2L/n
where:
λ = wavelength
L = length
n = harmonic (1,2,3,…)
equation of possible frequencies of string or open pipe:
ƒ = nv/2L
where: ƒ = frequency n = harmonic (1,2,3,...) v = wave speed L = length
equation that relates wavelength of standing wave and length of closed pipe:
λ = 4L/n
where:
λ = wavelength
L = length
n = harmonic (1,3,5,…)
intensity (I)
I = P/A
where:
P = power
A = area
focal point (ƒ)
ƒ = r / 2
where:
ƒ = focal point
r = radius of curvature
relationship between distances in geometrical optics:
1/ƒ = 1/o + 1/i = 2/r
where: ƒ = focal length o = distance between object and mirror i = distance between image and mirror r = radius of curvature
magnification (m)
m = -i/o
where:
m = magnification
i = distance between image and mirror
o = distance between object and mirror
ray diagrams for concave mirrors:
object is placed beyond F (focal point)
ray diagrams for concave mirrors:
object is placed at F (focal point)
ray diagrams for concave mirrors:
object is placed between F (focal point) and the mirror
ray diagram for convex mirrors:
index of refraction (n)
n = c/v
where:
n = index of refraction
c = speed of light in vacuum (3x10^8 m/s)
v = speed of light in medium
Snell’s law (law of refraction)
n(1) sin θ(1) = n(2) sin θ(2)
where:
n(1) and θ(1) refer to medium light comes from
n(2) and θ(2) refer to medium light enters
lensmaker’s equation
P = 1/ƒ = (n-1) (1/r(1) - 1/r(2))
where:
ƒ = focal length
n = index of refraction of lens material
r(1) and r(2) = radius of curvature of first and second lenses
power (P)
P = 1/ƒ
where:
ƒ = focal length
addition of multiple lens systems:
focal length-
power-
magnification-
1/f = 1/f(1) + 1/f(2) + 1/f(3) + ... + 1/f(n) P = P(1) + P(2) + P(3) + ... + P(n) m = m(1) x m(2) x m(3) x ... x m(n)
relationship of energy to frequency of light:
E = hƒ
where:
E = energy of photon of light
h = Planck’s constant (6.626x10^-34 J*s)
ƒ = frequency of light
maximum kinetic energy of ejected electron:
K(max) = hƒ - W
where:
h = Planck’s constant (6.626x10^-34 J*s)
ƒ = frequency of light
W = work function of metal in question
work function:
the minimum energy necessary to eject an electron from a given metal
W = h ƒ(T)
where:
h = Planck’s constant (6.626x10^-34 J*s)
ƒ(T) = threshold frequency
equivalence of matter and energy:
E = mc^2
where:
E = energy
m = mass
c = speed of light (3x10^8 m/s)
isotopic notation
elements are preceded by their atomic number (Z) as a subscript and mass number (A) as a superscript
where:
X = element
A = mass number (corresponds to number of protons plus number of neutrons)
Z = atomic number (corresponds to number of protons)
radioactive decay:
alpha (α) decay balanced equation:
radioactive decay:
beta-negative (β-) decay balanced equation:
radioactive decay:
beta-positive (β+) decay balanced equation:
radioactive decay:
gamma (𝛾) decay balanced equation:
radioactive decay:
electron capture balanced equation:
rate at which nuclei decay:
∆N/∆t = -λn
where:
∆N/∆t = rate at which nuclei decay
λ = decay constant
n = number of radioactive nuclei that have not yet decayed
exponential decay:
n = n(o) e^-λt
where:
n = number of radioactive nuclei that have not yet decayed
n(o) = number of undecayed nuclei at time t = 0
λ = decay constant
t = time
decay constant relation to half-life:
λ = ln 2 / T(1/2) = .693 / T(1/2)
where:
λ = decay constant
T(1/2) = half-life
rules of logarithms:
log(A) 1 = __
0
rules of logarithms:
log(A) A = __
1
rules of logarithms:
log A*B = __
log A + log B
rules of logarithms:
log A/B = __
log A - log B
rules of logarithms:
log A^B = __
B log A
rules of logarithms:
log 1/A = __
- log A
conversion between common and natural logarithms:
log x ≈ ln x / 2.303
estimating logarithms:
e.g. log of 7,426,135,420 –>
log (n * 10^m) ≈ m + 0.n
e.g. log of 7,426,135,420 –> 7.4 x 10^9 –> 9 + 0.74 = 9.74 (actual = 9.87)
Celcius to Farenheit:
F = 9/5 C + 32
