measurement
SI units, constants, vectors, graph work, uncertainty propagation, cross-topic relations.
units & prefixes · constants · vectors · calculus & graphs · measurement · uncertainty propagation · cross-topic relations
Units & prefixes
Derived units. \(\mathrm{N}=\mathrm{kg\,m\,s^{-2}}\), \(\mathrm{J}=\mathrm{N\,m}\), \(\mathrm{W}=\mathrm{J\,s^{-1}}\), \(\mathrm{Pa}=\mathrm{N\,m^{-2}}\), \(\mathrm{C}=\mathrm{A\,s}\), \(\mathrm{V}=\mathrm{J\,C^{-1}}\), \(\Omega=\mathrm{V\,A^{-1}}\), \(\mathrm{F}=\mathrm{C\,V^{-1}}\), \(\mathrm{T}=\mathrm{N\,A^{-1}\,m^{-1}}\), \(\mathrm{Wb}=\mathrm{T\,m^2}\), \(\mathrm{Hz}=\mathrm{s^{-1}}\).
Prefixes.
| Symbol | Value |
|---|---|
| \(\mathrm{T}\) | \(10^{12}\) |
| \(\mathrm{G}\) | \(10^{9}\) |
| \(\mathrm{M}\) | \(10^{6}\) |
| \(\mathrm{k}\) | \(10^{3}\) |
| \(\mathrm{c}\) | \(10^{-2}\) |
| \(\mathrm{m}\) | \(10^{-3}\) |
| \(\mu\) | \(10^{-6}\) |
| \(\mathrm{n}\) | \(10^{-9}\) |
| \(\mathrm{p}\) | \(10^{-12}\) |
Constants
Values. SI units unless stated.
| Symbol | Value |
|---|---|
| \(c\) | \(2.998\times10^{8}\) |
| \(g\) | \(9.81\) |
| \(G\) | \(6.67\times10^{-11}\) |
| \(h\) | \(6.626\times10^{-34}\) |
| \(\hbar\) | \(h/2\pi\) |
| \(e\) | \(1.602\times10^{-19}\) |
| \(k\) | \(1.381\times10^{-23}\) |
| \(N_A\) | \(6.022\times10^{23}\) |
| \(R\) | \(8.314\) |
| \(\sigma\) | \(5.67\times10^{-8}\) |
| \(b\) | \(2.90\times10^{-3}\) |
| \(\epsilon_0\) | \(8.85\times10^{-12}\) |
| \(\mu_0\) | \(4\pi\times10^{-7}\) |
| \(k_e\) | \((4\pi\epsilon_0)^{-1}\) |
| \(m_e\) | \(9.11\times10^{-31}\) |
| \(m_p\) | \(1.673\times10^{-27}\) |
| \(m_n\) | \(1.675\times10^{-27}\) |
| \(u\) | \(1.661\times10^{-27}\) |
| \(1\,\mathrm{eV}\) | \(1.602\times10^{-19}\ \mathrm{J}\) |
| \(1\,u\) | \(931.5\ \mathrm{MeV}\,c^{-2}\) |
Vectors
Components. \(\vv{A}=A_x\vv{i}+A_y\vv{j}\,(+A_z\vv{k})\), \(|\vv{A}|=(A_x^2+A_y^2+A_z^2)^{1/2}\); \(A_x=A\cos\theta\), \(A_y=A\sin\theta\).
Dot product. \(\vv{A}\cdot\vv{B}=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z\).
Cross product. \(|\vv{A}\times\vv{B}|=AB\sin\theta\); direction by right-hand rule.
Calculus & graphs
Derivatives & integrals. \(\dfrac{\dd x^n}{\dd x}=nx^{n-1}\), \(\int x^n\dd x=x^{n+1}/(n+1)+C\), \(\dfrac{\dd e^{kx}}{\dd x}=ke^{kx}\), \(\int e^{kx}\dd x=e^{kx}/k\).
Graph readings. Gradient \(m=\Delta y/\Delta x\); area under curve \(=\int y\dd x\).
Linearize. Aim for \(y=mx+c\); \(y=Ax^n\Rightarrow\ln y=\ln A+n\ln x\); \(y=Ae^{kx}\Rightarrow\ln y=\ln A+kx\).
Measurement
Statistics. \(\bar x=\dfrac1n\sum x_i\), \(s=\sqrt{\dfrac{\sum(x_i-\bar x)^2}{n-1}}\), \(s_{\bar x}=s/\sqrt n\).
Resolution uncertainty. Digital: \(\pm\) last digit; analogue: \(\pm\tfrac12\) smallest division. Repeated readings: \(\Delta x\approx(x_{\max}-x_{\min})/2\).
Forms. Absolute \(\Delta x\); fractional \(\Delta x/x\); percent \(100\,\Delta x/x\).
Uncertainty propagation
Sum/difference. \(z=a\pm b\Rightarrow\Delta z=\Delta a+\Delta b\).
Product/quotient. \(z=ab\) or \(a/b\Rightarrow\dfrac{\Delta z}{|z|}=\dfrac{\Delta a}{|a|}+\dfrac{\Delta b}{|b|}\).
Powers. \(z=ka^m b^n/c^p\Rightarrow\dfrac{\Delta z}{|z|}=|m|\dfrac{\Delta a}{|a|}+|n|\dfrac{\Delta b}{|b|}+|p|\dfrac{\Delta c}{|c|}\).
General function. \(z=f(x)\Rightarrow\Delta z\approx|f'(x)|\,\Delta x\).
Best-fit gradient. Use steepest/shallowest acceptable lines; \(\Delta m=(m_{\max}-m_{\min})/2\).
Cross-topic relations
Energy carriers. Mechanical \(K,U\); thermal \(Q,U\); electrical \(qV,\,Pt\); photon \(hf\); rest \(mc^2\).
Fields analogy. Gravity: \(F=GmM/r^2\), \(g=F/m\), \(V_g=-GM/r\), \(U=mV_g\). Electric: \(F=kqQ/r^2\), \(E=F/q\), \(V=kQ/r\), \(U=qV\). Uniform field work: \(\Delta U=-W_{\mathrm{field}}\), force \(=-\nabla U\).
Exponentials. Decay/discharge: \(y=y_0e^{-t/\tau}\), half time \(t_{1/2}=\tau\ln 2\). Growth/charging: \(y=y_\infty(1-e^{-t/\tau})\). Linear on semilog: \(\ln y=\ln y_0-t/\tau\).
Common rearrangements. From \(v^2/r=GM/r^2\): \(v=\sqrt{GM/r}\). From \(qvB=mv^2/r\): \(r=p/(qB)\). From \(qV=\tfrac12mv^2\): \(v=\sqrt{2qV/m}\). From \(P=IV\) at fixed \(P\): high \(V\) gives low \(I\) and low \(I^2R\) loss.
Proportionalities. \(F_g,F_e\propto r^{-2}\); \(V_g,V_e,U\propto r^{-1}\). Wave \(I\propto A^2\) and spherical \(I\propto r^{-2}\). Blackbody \(L\propto R^2T^4\); \(\lambda_{\max}\propto 1/T\). Gas at fixed \(n\): \(p\propto T/V\). SHM spring: \(T\propto\sqrt m\), \(T\propto 1/\sqrt k\); pendulum \(T\propto\sqrt L\).