Formulae list

Some of these formulae should be memorised during your higher level course but the list covers a great deal more than is needed for a simple extension to the foundation course. It is most important that you are sure of the meaning of all the symbols

Equations of motion

s = vt
v = u + at
v² = u² + 2as
s = ut + ½ at²
average vel. = [u + v]/2

Dynamics

Momentum M = mv
Impulse I = Ft
Newton's second law F = d(mv)/dt = ma
Impulse and momentum Ft = mv – mu
Kinetic energy k.e = ½ mv²
Potential energy p.e = mgh
Work work = Fs
Power Power = Fv

Statics

Weight (F) = mg
Pressure (P) = F/A
Pressure at a depth h in a liquid in a liquid (P) = hρg
Density (ρ) = m/V
Couple (C) = Fd
Upthrust (U) = vρg

Projectiles

Range (R) = u²sin2A/g
Maximum height (h) = u²sin2A/2g
Time of flight t = 2usinA/g

Motion in a circle

Angular velocity ω = θ/t
Linear and angular velocity v = rω
Time of rotation period T = 2πr/v = 2π/ω
Centripetal force F = mv²/r = mω²r

Rotational dynamics

Moment of inertia (I) = Σmr²
Angular momentum (M) = Iω
Rotational kinetic energy = ½ Iω²
Couple (C) = Iα
Work done W = Cθ

Simple harmonic motion

Acceleration a = -ω² x
Displacement x = rsin(ωt)
Velocity v = ±ω√(r² - x²)
Acceleration a = -ω²rsin(ωt)
Velocity v = ωr cos (ωt)
Kinetic energy = ½ mω²(r² - x²)
Potential energy = ½ mω²x²
Total energy E = ½ mω²r²

Gravitation

Kepler's third law T²/r³ = constant
Newton's law F = Gm₁m₂/d²
Potential energy p.e = - GmM/r
Kinetic energy k.e = +GmM/2r
Total energy E = - GmM/2r
Potential V_G = - GM/r
g_o and G g_o = GM/R²
g and g_o (r> R) g = g_oR²/r²
g and g_o (r < R) g = g_or/R
Escape velocity v = √[2Rg_o]

Elasticity

Stress   stress = F/A
Strain strain = e/L
Young modulus   E = F/LeA
Bulk modulus   K = Dp/(Dv/v)
Rigidity or shear modulus (G) G = [F/A]/θ
Potential energy stored = ½ Fe = ½ EAe²/L
Energy per unit volume = ½ stress x strain

Thermal expansion

F = EAαθ

Friction

Coefficient of friction (μ) F = μR

Viscosity

Coefficient of viscosity (η) F = hA x velocity gradient
Stokes' law F = 6πhrv
Poiseuille's formula Volume s^-1 = πhρgr⁴/8ηl

Surface tension

Capillary rise (h) T cosθ = hrρg/2
Excess pressure in air bubble p = 2T/r
Excess pressure in soap bubble p = 4T/r

Geometrical Optics

Refractive index (n) = sin i/sin r = real depth/apparent depth
Related to wave velocities (n) = c_m/c_v
Serial relation n₁sinθ₁ = n₂sinθ₂
Thin prism   d = (n – 1)A
Critical angle(c) n = 1/sin c
Lens formulae 1/u + 1/v = 1/f
Telescope magnification (m) = f_o/f_e
Angular magnification   M = - (D/f+ 1)
Resolving power f = 1.22λ/a

Physical Optics

Constructive interference path difference for a maximum = mλ
Destructive interference path difference for a minimum= (2m + 1)λ/2
Young's slits   ml = x_md/D
Newton's rings (dark ring viewed by reflection) mλ = r_m²/R
Thin film interference mλ = 2nt cos r
Diffraction grating (max)    mλ = e sinθ
Brewster's law (polarisation) tan p = n
Malus' law I = I_ocos²θ

Wave motion

Doppler effect Δλ = λv/c Δf = fv/c
Travelling wave y = a sin[ωt – kx] = a sin2π[t/T – x/λ]
Standing wave y = 2a cos[2πx/λ]sin[2πt/T]
Velocity of sound (v) = √[γP/ρ]
Frequency of stretched string f_o = (1/2L)√[T/m]^1/2
Fundamental frequency (closed tube) f_o = v/4L
Intensity of wave I = ka²
Beat frequency f = f₁ – f₂
Organ pipes:
Open pipe f = (m + 1)f_o
Closed at one end f = (2m + 1)f_o

Thermal Physics

Scale of temperature t/100 = (F_t – F₀)/(F₁₀₀ – F₀)
Linear expansivity (α) L_θ = L₀[1 + αθ]
Specific heat capacity (c) H = mcθ
Specific latent heat (L) H = mL
Electrical heating H = VIt
Density change r_θ = ρ₀[1+αθ]
Ideal gas equation PV = nRT
Isothermal change PV = constant
Adiabatic change PV^γ = constant
Charles's law V/T = constant
Conduction of heat dH/dt = - kA dθ/dx
Stefan-Boltzmann law E = σA[T⁴ – T_o⁴]
Wien's law λ_max = constant
First law of thermodynamics dU = dQ + dW
Work done in isothermal change dW = PdV
Kinetic theory equation PV = 1/3 m_nc²_rms
Mean square velocity (c_rms)² = v[u₁² + u₂² + …. + u_n²]/n

Electricity

Charge Q = It
Current I = nAve
Electrical energy = QV
Force on charge F = QE = QV/d
Ohm's law V = IR
Internal resistance E = I[R + r]
Resistivity ρ= RL/A
Temperature variation R_θ = R_o[1 + αθ]
Series resistance R = R₁ + R₂
Parallel resistance 1/R = 1/R₁ + 1/R₂
Power W = VI = I²R = V²/R

Electrostatics

Electric field strength E = - dV/dx
Force between point charges F = Q₁Q₂/[4πεd²]
Field due to point charge Q E = Q/[4εd²]
Potential V = W/Q_o
Potential due to charge Q V_E = Q/[4πεd]
Capacitance (C) = Q/V
Capacitance of a sphere (C) = 4πεr
Parallel-plate capacitor C = εA/d
Parallel capacitors      C = C₁ + C₂
Series capacitors      1/C = 1/C₁ + 1/C₂
Energy (E) stored by a capacitor = ½ CV² = ½ QV = ½ Q²/C
Capacitor discharge      V = V_oe^-t/RC
Capacitor charge      V = V_o[1 - e^-t/RC]

Electromagnetism

Couple on coil (C) = BANIsinθ
Field at centre of coil of N turns (B) = μ_oNI/2r
Field in solenoid (B) = μ_oNI/L
Field at end of long solenoid of N turns (B) = μ_oNI/2L
Helmholtz coils field (B) = 8μ_oNI/5[5]^1/2r
Field near straight wire (B) = μ_oI/2πr
Velocity of e.m. waves c = 1/(ε_oμ_o)^1/2

Electromagnetic induction

Self-inductance L = Nφ/I
Mutual inductance M = N_sφ_s/I_p
Induced e.m.f. (ε) = -L dI/dt
Induced e.m.f. (ε_s) = - MdI/dt
Induced e.m.f. in a rotating coil (ε) = BANωsinθ
Induced e.m.f.(Neumann's law) ε = - Ndφ/dt
Transformer      n_p/n_s = V_p/V_s and I_p/I_s = n_s/n_p
Root mean square current (I) I = i_o/2^1/2
Alternating current i = io sin(ωt)
Capacitative reactance X_c = 1/ωC
Inductive reactance X_L = ωL
Impedance (series RLC) Z = [R² + (X_L _ X_C)²]^1/2
Resonance condition for I     X_L = X_C

Electron Physics

Electrostatic f orce on electron F = eE
Electromagnetic force on electron F = Bev
Crossed fields     eE = Bev
Energy gain     E = eV
Kinetic energy      eV = ½ mv²
Circular orbit      Bev = mv²/r
Quantum energy      E = hf
Relativistic mass-energy relation      E = mc²
de Broglie equation λ = h/p = h/mv
Work function     W = hf_o
Einstein's p.e. equation     hf = hf_o + ½ mv²
Photoelectric effect     hf = eV

Nuclear Physics

Radioactive decay N = N_oe^-lt
A = A_o/2ⁿ Half-life T = ln2/λ
Serial relation λ₁N₁ = λ₂N₂
Nuclear radius (r) of nucleus of mass number A = r_oA^1/3

Relativity

m = m_o/[1-v²/c²]^½
L = L_o[1-v²/c²]^½
t = t_o/[1-v²/c²]^½
γ = 1/[1-v²/c²]^1/2 = [1-v²/c²]^-1/2
m = γm_o
L = L_o/γ
t = γt_o

 

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