Physics Equations (PHY 2130 / 2140)

Below is a compact equation sheet for Physics 1 students.
Each equation includes a quick guide to symbols.

Constants

  • $G = 6.67\times 10^{-11}\, \mathrm{N\,m^2/kg^2}$
  • $M_{\oplus} = 5.97\times 10^{24}\, \mathrm{kg}$
  • $R_{\oplus} = 6.38\times 10^{6}\, \mathrm{m}$
  • $g = 9.8\, \mathrm{N/kg} = 9.8\, \mathrm{m/s^2}$
  • $k_B = 1.38\times 10^{-23}\, \mathrm{J/K}$
  • $R = 8.31\, \mathrm{J/(mol\,K)}$
  • $e = 1.602\times 10^{-19}\, \mathrm{C}$
  • $k_c = 8.99\times 10^{9}\, \mathrm{N\,m^2/C^2}$
  • $\varepsilon_0 = 8.85\times 10^{-12}\, \mathrm{C^2/(N\,m^2)}$
  • $\mu_0 = 4\pi\times 10^{-7}\, \mathrm{T\,m/A}$

Vector / Trig Summary

Right triangle: \(\text{hyp}^2 = \text{opp}^2 + \text{adj}^2\)

Trig: \(\sin\theta = \frac{\text{opp}}{\text{hyp}},\qquad \cos\theta = \frac{\text{adj}}{\text{hyp}},\qquad \tan\theta = \frac{\text{opp}}{\text{adj}}\)

Symbols

  • $\theta$: angle
  • hyp/opp/adj: hypotenuse / opposite / adjacent

Vector components: \(\vec A = A_x \hat i + A_y \hat j\)

\[A_x = A\cos\theta,\qquad A_y = A\sin\theta\]

Symbols

  • $\vec A$: vector
  • $A$: magnitude
  • $A_x, A_y$: components
  • $\hat i,\hat j$: unit vectors

Unit 2 – Motion (Kinematics)

Position: \(\vec r = x\,\hat i + y\,\hat j\)

Displacement: \(\Delta\vec r = \vec r_f - \vec r_i\)

Average speed: \(v_{\text{avg}} = \frac{\text{distance}}{\Delta t}\)

Average velocity: \(\vec v_{\text{avg}} = \frac{\Delta\vec r}{\Delta t}\)

Average acceleration: \(\vec a_{\text{avg}} = \frac{\Delta\vec v}{\Delta t}\)

Constant-acceleration relations: \(v_f = v_i + a t\)

\[\Delta x = v_i t + \frac12 a t^2\] \[v_f^2 = v_i^2 + 2a\Delta x\]

Symbols

  • $x,y$: position coordinates (m)
  • $\vec r_i,\vec r_f$: initial/final position
  • $v_i,v_f$: initial/final velocity (m/s)
  • $a$: acceleration (m/s$^2$)
  • $t,\Delta t$: time (s)
  • $\Delta x$: displacement (m)

Unit 3 – Forces & Newton’s Laws

Net force: \(\vec F_{\text{net}}=\sum \vec F\)

Newton’s 2nd law: \(\vec F_{\text{net}}=m\vec a\)

Weight: \(W = mg\)

Hooke’s law: \(F_s = k\Delta x\)

Friction: \(f_s \le \mu_s N,\qquad f_k=\mu_k N\)

Impulse: \(\vec I = \vec F\,\Delta t = m\Delta\vec v\)

Symbols

  • $m$: mass (kg)
  • $N$: normal force (N)
  • $k$: spring constant (N/m)
  • $\mu_s,\mu_k$: static/kinetic friction coefficients
  • $\vec I$: impulse (N·s)

Unit 4 – Solids & Fluids

Density: \(\rho = \frac{m}{V}\)

Pressure: \(P=\frac{F}{A}\)

Hydrostatic pressure: \(P = P_0 + \rho g d\)

Buoyant force: \(F_B = \rho_f V_f g\)

Surface tension pressure: \(\Delta P = \frac{2\gamma}{r}\)

Continuity: \(Q = Av = \text{constant}\)

Poiseuille’s law: \(Q = \frac{\pi R^4}{8\mu L}\Delta P\)

Bernoulli: \(P+\frac12\rho v^2+\rho g y = \text{constant}\)

Symbols

  • $\rho$: density (kg/m$^3$)
  • $P$: pressure (Pa)
  • $P_0$: surface pressure
  • $d$: depth (m)
  • $F_B$: buoyant force (N)
  • $\rho_f$: fluid density
  • $V_f$: displaced volume
  • $\gamma$: surface tension
  • $Q$: flow rate (m$^3$/s)
  • $R$: tube radius, $L$: tube length
  • $\mu$: viscosity

Unit 5 – Energy & Work

Work: \(W = F_{\parallel} d\)

Kinetic energy: \(K = \frac12 mv^2\)

Gravitational potential energy: \(U_g = mgy\)

Spring potential energy: \(U_s=\frac12 k(\Delta x)^2\)

Total mechanical energy: \(E_{\text{tot}} = K + U + E_{\text{th}} + \dots\)

Power: \(P=\frac{\Delta E}{\Delta t}\)

Symbols

  • $W$: work (J)
  • $K$: kinetic energy (J)
  • $U_g,U_s$: potential energies (J)
  • $P$: power (W)

Unit 6 – Thermodynamics

Temperature conversion: \(T(K)=T(^{\circ}C)+273.15\)

Heat: \(Q=mc\Delta T\)

Ideal gas law: \(PV = Nk_BT = nRT\)

Average kinetic energy (ideal gas): \(K_{\text{avg}}=\frac32 k_B T\)

RMS speed: \(v_{\text{rms}}=\sqrt{\frac{3k_BT}{m}}\)

First law: \(\Delta U = W + Q\)

Symbols

  • $Q$: heat (J)
  • $c$: specific heat (J/kg·K)
  • $n$: moles
  • $N$: number of molecules
  • $U$: internal energy (J)

Unit 7 – Electricity & Magnetism

Coulomb’s Law

\(F_e = k_c\frac{|q_1 q_2|}{r^2}\)

Symbols

  • $F_e$: electric force (N)
  • $k_c$: Coulomb constant
  • $q_1,q_2$: charges (C)
  • $r$: separation (m)

Electric Field

\(E = k_c\frac{|q|}{r^2}\)

\[\vec F = q\vec E\]

Symbols

  • $E$: electric field (N/C or V/m)
  • $q$: charge (C)
  • $\vec F$: force on charge (N)

Electric Potential & Energy

\(V = k_c\frac{q}{r}\)

\[U_e = k_c\frac{q_1 q_2}{r}\] \[\Delta U = q\Delta V\]

Symbols

  • $V$: electric potential (V)
  • $U_e$: electric potential energy (J)
  • $\Delta V$: potential difference

Capacitance

\(C=\frac{Q}{V}\)

\[C=\varepsilon_0\frac{A}{d}\] \[U=\frac12 CV^2=\frac12 QV=\frac{Q^2}{2C}\]

Symbols

  • $C$: capacitance (F)
  • $Q$: charge stored (C)
  • $V$: voltage (V)
  • $\varepsilon_0$: permittivity of free space
  • $A$: plate area (m$^2$)
  • $d$: separation (m)
  • $U$: stored energy (J)

Current & Ohm’s Law

\(I=\frac{\Delta Q}{\Delta t}\)

\[V = IR\]

Power: \(P = IV = I^2R = \frac{V^2}{R}\)

Resistors in series: \(R_{\text{eq}}=R_1+R_2+\cdots\)

Resistors in parallel: \(\frac1{R_{\text{eq}}}=\frac1{R_1}+\frac1{R_2}+\cdots\)

Symbols

  • $I$: current (A)
  • $R$: resistance (Ω)
  • $P$: power (W)

Magnetic Force

Moving charge: \(F_B=qvB\sin\theta\)

Current-carrying wire: \(F=ILB\sin\theta\)

Symbols

  • $B$: magnetic field (T)
  • $\theta$: angle between motion/current and $B$

Magnetic Field from Currents

Long straight wire: \(B=\frac{\mu_0 I}{2\pi r}\)

Solenoid: \(B=\mu_0 nI\)

Symbols

  • $\mu_0$: permeability of free space
  • $n$: turns per unit length
  • $r$: distance from wire

Magnetic Flux & Induction

\(\Phi_B = BA\cos\theta\)

Faraday’s Law: \(\varepsilon = -\frac{d\Phi_B}{dt}\)

Symbols

  • $\Phi_B$: magnetic flux (Wb)
  • $\varepsilon$: induced emf (V)
  • $A$: area of loop (m$^2$)

(New) Capacitors in Series/Parallel

Series: \(\frac{1}{C_{\text{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots\)

Parallel: \(C_{\text{eq}}=C_1+C_2+\cdots\)

Symbols

  • $C_{\text{eq}}$: equivalent capacitance (F)

(New) Lorentz Force (vector form)

\[\vec F = q\vec E + q\,\vec v\times \vec B\]

Symbols

  • $\vec v$: velocity of charge (m/s)
  • $\times$: cross product

Unit 8 – Diffusion, Brownian Motion & Terminal Velocity (Labs)

Brownian Motion / Diffusion

Mean-square displacement:

1D: \(x_{\text{rms}}^2 = 2Dt\)

2D: \(r_{\text{rms}}^2 = 4Dt\)

3D: \(r_{\text{rms}}^2 = 6Dt\)

Diffusion flux (Fick’s law): \(J = -D\frac{\Delta n}{\Delta x}\)

Stokes-Einstein relation: \(D = \frac{k_B T}{6\pi\mu r}\)

Symbols

  • $D$: diffusion constant (m$^2$/s)
  • $t$: time (s)
  • $J$: diffusion flux
  • $n$: concentration
  • $\mu$: viscosity (Pa·s)
  • $r$: particle radius (m)
  • $T$: temperature (K)

Viscous Drag (Stokes’ Law)

Force on a small sphere moving in a viscous fluid: \(F_{\text{viscous}} = 6\pi\mu r v\)

Symbols

  • $F_{\text{viscous}}$: drag force (N)
  • $v$: speed (m/s)

Terminal Velocity (falling sphere)

When drag balances effective weight:

\[v_t = \frac{2r^2(\rho_s-\rho_f)g}{9\mu}\]

Symbols

  • $v_t$: terminal velocity (m/s)
  • $\rho_s$: sphere density (kg/m$^3$)
  • $\rho_f$: fluid density (kg/m$^3$)

Vesicle Transport (motor proteins)

Velocity-ATP relation: \(v = R\,s\)

Symbols

  • $v$: vesicle speed (m/s)
  • $R$: ATP hydrolysis rate (cycles/s)
  • $s$: motor step size (m)
    • kinesin: $s \approx 8\,\mathrm{nm}$
    • myosin V: $s \approx 36\,\mathrm{nm}$

Useful pixel → SI conversions (ImageJ labs)

If your tracking gives values in micrometers:

\[1\ \mu\text{m} = 10^{-6}\ \text{m}\]

If tracking gives pixels and your calibration is:

\[1\ \text{pixel} \approx 0.193\ \mu\text{m}\]

then

\[v(\text{m/s}) = v(\text{pixels/s})\times 0.193\times 10^{-6}\]

and

\[r(\text{m}) = r(\text{pixels})\times 0.193\times 10^{-6}\]