Formulas Flashcards
Arc length
s = rθ
(s) arc length [m]
(r) radius of circle [m]
(θ) angular displacement [rad]
Angular speed
ω = ∆θ / ∆t
(ω) angular speed [rad s⁻¹]
(θ) angular displacement [rad]
(t) time taken [s]
Angular speed equation for one revolution
ω = 2π / T
(ω) angular speed [rad s⁻¹]
(2π) angle rotated through by object in one revolution [rad]
(T) period/time to make one revolution [s]
Equation relating speed and angular speed
v = ωr
(v) speed [ms⁻¹]
(ω) angular speed [rad s⁻¹]
(r) radius [m]
Acceleration equations in circular motion (2)
a = v²/r or a = ω²r
(a) acceleration [ms⁻²]
(v) speed [ms⁻¹]
(r) radius [m]
(ω) angular velocity [rad s⁻¹]
Combining Newton’s second law and acceleration equations in circular motion
F = ma
a = v²/r or a = ω²r
∴
Fc = mv²/r or mω²r
(F) force [N]
(a) acceleration [ms⁻²]
(v) speed [ms⁻¹]
(r) radius [m]
(ω) angular velocity [rad s⁻¹]
(Fc) centripetal force [N]
(m) mass [kg]
Calculating orbital speed
F = ma
F = mv²/r
∴
ma = mv²/r
a = v²/r
(F) force [N]
(m) mass [kg]
(a) acceleration [ms⁻²]
(v) speed [ms⁻¹]
(r) radius [m]
Angular frequency equation
ω = 2πf
(ω) angular frequency [rad s⁻¹]
(f) frequency [Hz]
Equations of s.h.m
x = x₀sin ωt or x = x₀cos ωt
sine version used when x (displacement curve) = 0 at t = 0
cosine version used when x (displacement curve) = x₀ at t = 0
(x) displacement [m]
(x₀) max displacement [m]
(ω) angular frequency [rad s⁻¹]
(t) time [s]
Acceleration and displacement formula s.h.m
a = -ω²x
(a) acceleration [ms⁻²]
(ω) angular frequency [rad s⁻¹]
(x) displacement [m]
Equation for velocity in s.h.m
v = v₀cos ωt
cosine version used as the velocity is at a maximum when t = 0
(v) velocity [ms⁻¹]
(v₀) max velocity [ms⁻¹]
(ω) angular frequency [rad s⁻¹]
(t) time [s]
Speed of an oscillator (2)
v = (±)ω √x₀² - x²
v₀ = ωx₀
(v) velocity [ms⁻¹]
(ω) angular frequency [rad s⁻¹]
(x₀) max displacement [m]
(x) displacement [m]
(v₀) max velocity [ms⁻¹]
Total energy of a system undergoing s.h.m
E = 1/2mv₀² and v₀ = ωx₀
∴
E = 1/2mω²x₀²
(E) maximum kinetic energy and total energy of system [J]
(m) mass [kg]
(v₀) max velocity [ms⁻¹]
(ω) angular frequency [rad s⁻¹]
(x₀) max displacement [m]
Internal energy
∆U = q + w
(U) increase in internal energy [J]
(q) energy supplied by heating [J]
(w) work done on the system [J]
Work done when the volume of a gas changes at constant pressure
F = p x A
W = F x d
∴
W = p∆V
(F) force [N]
(p) pressure [kgm⁻³]
(A) area [m²]
(W) work done [J]
(V) volume [m³]
Note:
true if gas is expanding against pressure of atmosphere which changes only very slowly
Conversion between celsius and kelvin
θºC = T (K) - 273.15
T(K) = θºC + 273.15
(θ) temperature in ºC
(T) temperature in K
Specific heat capacity (2)
E = mc∆θ or Q = mc∆T
or
specific heat capacity = energy supplied / mass temperature change
(E) energy supplied [J]
(m) mass [kg]
(c) specific heat capacity [Jkg⁻¹ºC⁻¹]
(θ) temperature [ºC]
(Q) thermal energy supplied [J]
(T) temperature [K]
Energy formula involving power
E = p x t
(E) energy [J]
(p) power [W]
(t) time [s]
Specific latent heat (2)
E = mL or Q = mL
or
specific latent heat = rate of supply of energy / rate of loss of mass
(E) energy required to melt or vapourise [J]
(m) mass [kg]
(L) specific latent heat [Jkg⁻¹]
(Q) thermal energy required to melt or vapourise [J]
Number of atoms in specific mass of a substance
number of atoms in a substance
specific mass of substance / mass of a single atom of that substance
Number of moles (2)
For a specific mass of a substance:
number of atoms in a specific weight of the substance / avogadro constant
Given molar mass:
mass / molar mass
Boyle’s law
pV = constant
p₁V₁ = p₂V₂
(p) pressure [kgm⁻³]
(V) volume [m³]
Charles’ law
V/T = constant
V₁/T₁ = V₂/T₂
(V) volume [m³]
(T) temperature [K]
Key equation for fixed mass of gas
pV/T = constant
p₁V₁/T₁ = p₂V₂/T₂
(p) pressure [kgm⁻³]
(V) volume [m³]
(T) temperature [K]
Equations of state (2)
pV = nRT or pV = NkT
(p) pressure [kgm⁻³]
(V) volume [m³]
(n) number of moles [mol]
(R) universal molar gas constant [Jmol⁻¹K⁻¹]
(T) temperature [K]
(N) number of molecules [mol]
(k) Boltzmann constant [JK⁻¹]
Number of molecules
number of molecules = number of moles x Avogadro’s constant
Pressure formula derivation
F = ∆mv/t
P = F/A
(F) force [N]
(∆mv) change in momentum [kgms⁻¹]
(p) pressure [kgm⁻³]
(A) area [m²]
Pressure of an ideal gas (3)
p = 1/3 (Nm/V) < c >²
or
pV = 1/3Nm< c >²
Nm/V is equal to the density of a gas
∴
p = 1/3 p < c >²
(p) pressure [kgm⁻³]
(N) number of molecules [mol]
(m) mass [kg]
(V) volume [m³]
(< c >²) mean square speed of molecules [(m/s)²]
Boltzmann constant equation
k = R/Nᴀ
(k) Boltzmann constant
(R) universal molar gas constant [Jmol⁻¹K⁻¹]
(Nᴀ) Avogadro’s constant
Kinetic energy formula from Boltzmann constant
1/2m< c> ² = 3/2kT
(m) mass [kg]
(< c >²) mean square speed of molecules [(m/s)²]
(k) Boltzmann constant
(T) temperature [K]
Root-mean-square-speed
cᵣ.ₘ.ₛ = √< c >²
(cᵣ.ₘ.ₛ) root-mean-square-speed [ms⁻¹]
(< c >²) mean square speed of
Newton’s law of gravitation
F = [ Gm₁m₂ ] / r²
(F) force [N]
(G) gravitational constant [Nm²kg⁻²]
(m) mass [kg]
(r) radius / centre-to-centre separation [m]
Gravitational field due to a point mass
g = F / m
F = [ Gm₁m₂ ] / r²
∴
g = GM/r²
(g) gravitational field strength [Nkg⁻¹]
(F) force [N]
(m) mass [kg]
(G) gravitational constant [Nm²kg⁻²]
(r) radius * of larger object / centre-to-centre separation / distance from mass [m]
(M) mass [kg] * of larger object
negative sign may be omitted
Gravitational potential (2)
Φ = - GM/r
∆Φ = - GM (1/r₁ - 1/r₂)
(Φ) gravitational potential [Jkg⁻¹]
(G) gravitational constant [Nm²kg⁻²]
(M) mass [kg]
(r) distance from mass [m]
Orbit speed
F = mv²/r and F = [ Gm₁m₂ ] / r²
∴
v² = GM / r
(F) force [N]
(m) mass [kg]
(v) speed [ms⁻¹]
(r) radius [m]
(G) gravitational constant [Nm²kg⁻²]
(r) radius / centre-to-centre separation [m]
Orbital period
v = 2πr/T
∴
v² = (4π²r²/T²) = GM / r
∴
T² = (4π²/GM)r³
(v) speed [ms⁻¹]
(r) radius [m]
(T) period [s]
(G) gravitational constant [Nm²kg⁻²]
(M) mass [kg]
Electric field strength
E = F/Q
(E) electric field strength [NC⁻¹]
(F) force on the charge [N]
(Q) charge [C]
Strength of a uniform field between two parallel metal plates
E = ∆V/∆d
(E) electric field strength [Vm⁻¹]
(V) voltage [V]
(d) separation [m]
Force on a charge
F = QE and F = -QV/d
∴
F = eV/d
(F) force on the charge [N]
(Q) charge [C]
(E) electric field strength [NC⁻¹]
(V) voltage [V]
(d) separation [m]
(e) electron with charge -e [e]
Coulomb’s law (3)
F = [ kQ₁Q₂ ] /r² and k = 1 / 4πε₀
∴
F = Q₁Q₂ / 4πε₀r²
(F) force between 2 charges [N]
(k) permittivity of free space [Fm⁻¹]
(Q) charge [C]
(r) distance between centres [m]
Electric field strength for a radial field
E = Q / 4πε₀r²
(E) electric field strength due to a point charge [NC⁻¹]
(Q) charge [C]
(ε₀) electrical constant [8.85 x 10⁻¹² Fm⁻¹]
(r) distance from the point [m]
Work done in moving charge
W = QV
(W) work done in moving charge [J]
(Q) charge [C]
(V) voltage [V]
Electric potential in a radial field due to a point charge
V = Q / 4πε₀r
(V) electric potential [V]
(Q) charge [C]
(ε₀) electrical constant [8.85 x 10⁻¹² Fm⁻¹]
(r) distance from the point [m]
Potential energy of a pair of point charges
Eₚ = Qq / 4πε₀r
(Eₚ) potential energy of the pair of point charges [J or eV]
(Q) point charge [C]
(q) point charge [C]
(ε₀) electrical constant [8.85 x 10⁻¹² Fm⁻¹]
(r) distance between the point charges [m]
Potential difference between 2 points from a charge
∆V = Q / 4πε₀[1/r₁ - 1/r₂]
(V) electric potential [V]
(Q) charge [C]
(ε₀) electrical constant [8.85 x 10⁻¹² Fm⁻¹]
(r) distance from the point [m]
Capacitance
C = Q/V
(C) capacitance [F]
(Q) magnitude of charge on each of the capacitor’s plates [C]
(V) potential difference across the capacitor [V]
Work done in charging up a capacitor
W = 1/2 QV
W = 1/2 CV²
W = 1/2 Q²/C
(W) work done by charging a capacitor [J]
(Q) charge [C]
(V) voltage / potential difference [V]
(C) capacitance [F]
Capacitors in parallel including derivation
Cₜₒₜₐₗ = C₁ + C₂ + C₃
(C) capacitance [F]
Derivation:
Q = Q₁ + Q₂ = C₁V + C₂V
Q = (C₁ + C₂)V
∴
Cₜₒₜₐₗ = C₁ + C₂ + C₃ …
Capacitors in series including derivation
1/Cₜₒₜₐₗ = 1/C₁ + 1/C₂ + 1/C₃
(C) capacitance [F]
Derivation:
V₂ = Q / C₁ and V₂ = Q / C₂
V = Q / Cₜₒₜₐₗ
V = V₁ + V₂
Q/Cₜₒₜₐₗ + Q/C₁ + Q/C₂
∴
1/Cₜₒₜₐₗ = 1/C₁ + 1/C₂ + 1/C₃ …
Capacitance of isolated bodies (2)
for conducting spheres
V = [1/4πε₀] [Q/r]
C = Q / V
∴
C = 4πε₀r
(V) voltage / potential difference [V]
(ε₀) electrical constant [8.85 x 10⁻¹² Fm⁻¹]
(Q) charge [C]
(r) radius [m]
(C) capacitance [F]
Time constant for a capacitor discharging
τ = RC
(τ) time constant [s]
(R) resistance (Ω)
(C) capacitance [F]
Exponential decay of charge on a capacitor (3)
I = I₀ exp (-[t/RC])
Q = Q₀ exp (-[t/RC])
V = V₀ exp (-[t/RC])
(I) current [A]
(I₀) initial current [A]
(t) time [s]
(R) resistance (Ω)
(C) capacitance [F]
(Q) charge [C]
(Q₀) initial charge [C]
(V) p.d [V]
(V₀) p.d [V]
exp means take ln from both sides,
side with exp just becomes the fraction {-[t/RC]}, and the other side becomes ln(__).
Force on the conductor (only when the conductor is at right-angles to the magnetic field)
F = BIL
(F) force on conductor [N]
(B) magnetic flux density of uniform field [T]
(I) current current in conductor [A]
(L) length of conductor in uniform magnetic field [m]
Force on a current carrying conductor
F = BIL sinθ
(F) force on conductor [N]
(B) magnetic flux density of uniform field [T]
(I) current current in conductor [A]
(L) length of conductor in uniform magnetic field [m]
Magnetic force experienced by a charged particle
F = BQv sinθ
(F) magnetic force [N]
(B) magnetic flux density of uniform field [T]
(Q) charge on the particle [C]
(v) velocity of particle [ms⁻¹]
Orbiting charged particles
Fc = mv²/r and BQv = mv²/r
∴
r = mv/BQ and p = BQ*r
(Fc) centripetal force [N]
(m) mass [kg]
(v) speed [ms⁻¹]
(r) radius [m]
(B) magnetic flux density of uniform field [T]
(Q) charge on the particle [C]
(v) velocity of particle [ms⁻¹]
(p) momentum [kgms⁻¹]
Q* charge can be replaced with e, if the charged particle is an electron
Charge to mass ratio of electron
e V𝒸ₐ = 1/2mₑv² and r = mₑv/Be
∴
e / mₑ = 2V𝒸ₐ / r²B²
(e) elementary charge [C]
(V𝒸ₐ) p.d between the cathode and the anode [V]
(m) mass [m]
(v) velocity [ms⁻¹]
(r) radius of orbit [m]
(B) magnetic flux density field [T]
Formula combining magnetic and electric force
eE = Bev
∴
v = E/B
E = V/d
∴
v = V/Bd
(e) elementary charge [C]
(E) electric field strength [NC⁻¹]
(B) magnetic flux density field [T]
(v) velocity [ms⁻¹]
(V) voltage [V]
(d) separation [m]
Hall voltage equation including derivation
Vʜ = BI / ntq
(Vʜ) hall voltage [V]
(B) magnetic flux density of field [T]
(I) current current in conductor [A]
(n) number density of charge carriers
(t) thickness of slice [m]
(q) charge of an individual charge carrier [C]
Derivation:
eE = Bev
eVʜ / d = Bev
eVʜ / d = BeI/nAe
Vʜ = BId/nAe
A = d x t
∴
Vʜ = BI / ntq
Magnetic flux linkage (2)
Magnetic flux linkage =
NΦ
or
BANcosθ
(N) number of turns for coil
(Φ) magnetic flux [Wb]
(B) magnetic flux density [T]
(A) cross-sectional area [m²]
(θ) angle between normal to the area and magnetic field [º]
Induced electromagnetic force (2)
E = - (∆(NΦ) / ∆t)
(E) magnitude of induced e.m.f [V]
(N) number of turns for coil
(Φ) magnetic flux [Wb]
(t) time [s]
(-) present due to Lenz’s law, necessary to emphasise principle of conservation of energy
E = BLv
(E) magnitude of induced e.m.f [V]
(B) magnetic flux density [T]
(L) length of wire [m]
(v) speed of wire [ms⁻¹]
Transformer formulas (3)
Vp / Vs = Np / Ns
Ps = Pp hence
Vp x Ip = Vs x Is
(V) voltage [V]
(p) primary coil
(P) power (W)
(s) secondary coil
(N) number of turns
(I) current [A]
Alternating current
I = I₀ sin ωt
(I) current at time t [A]
(I₀) peak current [A]
(ω) angular frequency of supply [rad s⁻¹]
(t) time [s]
*calculator must be in radians
Alternating voltages
V = V₀ sin ωt
(V) voltage at time t [V]
(V₀) peak voltage [V]
(ω) angular frequency of supply [rad s⁻¹]
(t) time [s]
*calculator must be in radians
Root-mean-square (r.m.s) value (2)
Iᵣ.ₘ.ₛ = I₀ / √2
Vᵣ.ₘ.ₛ = V₀ / √2
(Iᵣ.ₘ.ₛ) root-mean-square value of current [A]
(I₀) peak (maximum) current [A]
(Vᵣ.ₘ.ₛ) root-mean-square value of voltage [V]
(V₀) peak (maximum) voltage [V]
Power formulas (4)
P = VI
P = I²R
P = V²/R
Pₘₐₓ = Pₐᵥ𝓰 x 2
(P) power [W]
(V) voltage [V]
(I) current [A]
(Pₘₐₓ) maximum power [W]
(Pₐᵥ𝓰) average power [W]
Einstein relation (2)
E = hf and E = hc/λ
(E) energy of a photon [J]
(h) Planck’s constant [eV]
(f) frequency [Hz]
(c) wave speed [ms⁻¹]
(λ) wavelength [m]
Speed of any type of charged particle
v = √(2eV/m) or v = √(2Eᴋ/m)
(v) electron speed [ms⁻¹]
(e) electron charge [C]
(V) voltage [V]
(m) mass of particle [kg]
(Eᴋ) kinetic energy [J]
Einstein’s photoelectric equation
E = hc/λ = hf = Φ + 1/2mvₘₐₓ²
(E) energy of a photon [J]
(h) Planck’s constant [eV]
(f) frequency [Hz]
(c) wave speed [ms⁻¹]
(λ) wavelength [m]
(Φ) work function of the metal [J or eV]
(1/2mvₘₐₓ²) maximum kinetic energy of emitted photoelectron [J]
Equations when incident radiation frequency equals threshold frequency (3)
hf₀ = Φ
∴
f₀ = Φ/h
∴
λ₀ = hc/Φ
(h) Planck’s constant [Js]
(f₀) threshold frequency [Hz]
(Φ) work function [J or eV]
(λ₀) threshold wavelength [m]
(c) wave speed [ms⁻¹]
Momentum of a photon
p = E/c
(p) momentum [kgms⁻¹]
(E) energy of the photon [J]
(c) photon speed [ms⁻¹]
The energy of a photon, absorbed or emitted, as a result of an electron making a transition between two energy levels E₁ and E₂
hf = E₁ - E₂
hc/λ = E₁ - E₂
(h) Planck’s constant [Js]
(f) frequency [Hz]
(c) wave speed [ms⁻¹]
(E) energy levels [J or eV]
de Broglie wavelength equation
λ = h/p
(λ) wavelength [m]
(h) Planck’s constant [Js]
(p) momentum [kgms⁻¹]
Finding wavelength using angle of separation
λ = 2d sinθ
(λ) wavelength [m]
(d) spacing of layers [m]
(θ) angle of diffraction [º]
Einstein’s mass energy equation
E = mc²
(E) energy [J]
(m) mass [kg]
(c) speed of light [ms⁻¹]
Activity (2)
A = (-)λN = ∆N/∆t
(A) activity [Bq]
(λ) decay constant [s⁻¹]
(N) number of undecayed nuclei
(t) time [s]
Radioactive decay formula
x = x₀ e⁻*ᵗ
(x) activity [Bq]
(x₀) activity at time t = 0 [Bq]
(*) decay constant (λ) [s⁻¹]
(t) time [s]
Half-life and decay constant relationship
λ = ln2/t₀.₅
= 0.693 / t₀.₅
(λ) decay constant [s⁻¹]
(t) time [s]
Attenuation of x-rays as they pass through a uniform material
I = I₀ e⁻*ˣ
(I) transmitted intensity [Wm⁻²]
(I₀) initial intensity [Wm⁻²]
(*) attenuation coefficient (µ) [m⁻¹]
(x) thickness of the material [m]
Acoustic impedance
Z = ρc
(Z) acoustic impedance [kgm⁻²s⁻¹]
(ρ) density [kgm⁻³]
(c) speed of sound [ms⁻¹]
Intensity reflection fraction of the boundary between two materials
Iᵣ/I₀ = [ (Z₁ - Z₂) / (Z₁ + Z₂) ]²
(Iᵣ) reflected intensity [Wm⁻²]
(I₀) incident intensity [Wm⁻²]
(Z) acoustic impedances of materials [kgm⁻²s⁻¹]
Attenuation of ultrasound
I = I₀ e⁻*ˣ
(I) transmitted intensity [Wm⁻²]
(I₀) initial intensity [Wm⁻²]
(*) absorption coefficient (a) [m⁻¹]
(x) thickness of the material [m]
A-scan formulas (2)
thickness of bone = distance travelled by ultrasound / 2
= c∆t / 2
(c) speed of ultrasound [ms⁻¹]
(t) time interval between pulses [s]
Number of photons
energy available / energy of a photon
Radiant flux intensity
F = L / 4πd²
(F) radiant flux intensity [Wm⁻²]
(L) luminosity, power of star, [W]
(4πd²) surface area of sphere [m²]
- (d) diameter [m]
Hubble’s law
v = H₀d
(v) speed [ms⁻¹]
(H₀) Hubble constant [s⁻¹]
(d) distance of the galaxy [m]
Doppler redshift
∆λ / λ
≈
∆f / f
≈
v / c
(λ) wavelength of the electromagnetic waves from the source [m]
(f) frequency of the electromagnetic waves from the source
(v) recession speed of source [s⁻¹]
(c) speed of light in vacuum [ms⁻¹]
