Physics olympiad roadmap.

A rigid body rotating at angular speed $\omega$ about a fixed axis has every point moving in a circle. Adding $\tfrac12 m v^2$ over all its pieces gives rotational kinetic energy $\tfrac12 I\omega^2$, where the moment of inertia is $I = \sum m_i r_i^2$ ($r_i$ is the distance from the axis). $I$ plays the part of mass for rotation.

The rotational versions of Newton's laws are $\tau = I\alpha$ and $L = I\omega$; with no external torque, $L$ is conserved (the spinning skater pulling in her arms).

Values to know (mass $M$):

• Thin rod, length $\ell$, about its centre: $\tfrac{1}{12}M\ell^2$; about one end: $\tfrac13 M\ell^2$.
• Ring or thin hoop, radius $R$: $MR^2$. Solid disc or cylinder: $\tfrac12 MR^2$. Solid sphere: $\tfrac25 MR^2$.
• Parallel axis theorem: $I = I_{\mathrm{cm}} + Md^2$ for an axis a distance $d$ from a parallel axis through the centre of mass.

Rolling without slipping means the contact point is momentarily at rest, so $v = \omega R$. The total kinetic energy is then $\tfrac12 Mv^2 + \tfrac12 I\omega^2$. Static friction does no work on a rolling body, so energy conservation still works.

Newton's laws hold only in inertial frames. You can still work in an accelerating frame if you add an inertial force to every body.

• Frame with acceleration $\vec a_0$: add $-m\vec a_0$ to each mass. It acts like extra gravity, so the effective gravity is $\vec g_{\mathrm{eff}} = \vec g - \vec a_0$.
• Frame rotating at steady $\omega$: add the centrifugal force $m\omega^2 r$ outwards from the axis, and, for bodies moving in that frame, the Coriolis force $-2m\,\vec\omega\times\vec v$.

The method: pick the frame in which the body you care about is at rest or moves simply, draw the free-body diagram including the inertial forces, then use ordinary statics or $F = ma$. The centrifugal force can be given a potential energy $-\tfrac12 m\omega^2 r^2$, so a bead on a spinning wire sits at a minimum of $U_{\mathrm{grav}} - \tfrac12 m\omega^2 r^2$. The Coriolis force does no work because it is always perpendicular to the velocity.

Two checks: inertial forces have no reaction partner, and you must never add them when working in the ground frame as well.

The first law says energy is conserved; the second law says which way heat flows. The tool is entropy $S$. When a body at temperature $T$ takes in a small amount of heat $\mathrm{d}Q$ reversibly, its entropy rises by $\mathrm{d}S = \mathrm{d}Q/T$. Entropy is a state variable, so to find $\Delta S$ for any change you may invent a reversible path with the same start and end.

The second law: the total entropy of an isolated system never decreases. It stays constant only for reversible processes.

• Heat $Q$ at constant $T$ (melting, a large reservoir): $\Delta S = Q/T$.
• Heating a body of heat capacity $C$: $\Delta S = C\ln(T_2/T_1)$.
• Ideal gas: $\Delta S = nC_V\ln(T_2/T_1) + nR\ln(V_2/V_1)$.
• Carnot limit: an engine between $T_h$ and $T_c$ has efficiency at most $1 - T_c/T_h$. A refrigerator has coefficient of performance at most $T_c/(T_h - T_c)$.

The Carnot limit follows directly: in one cycle the engine's entropy is unchanged, the hot reservoir loses $Q_h/T_h$ and the cold one gains $Q_c/T_c$, and the total cannot fall, so $Q_c/Q_h \ge T_c/T_h$.

In a system at temperature $T$, the chance of finding a particle in a particular state of energy $E$ is proportional to the Boltzmann factor $e^{-E/kT}$, where $k = 1.38\times10^{-23}\ \mathrm{J\,K^{-1}}$. States much higher than $kT$ above the ground state are rarely occupied. At room temperature $kT \approx 0.025$ eV.

Results to know:

• Two levels with energy gap $\Delta E$ and equal degeneracy: $N_2/N_1 = e^{-\Delta E/kT}$.
• Isothermal atmosphere: the number density falls as $n(h) = n_0e^{-mgh/kT}$, with scale height $H = kT/mg$.
• Equipartition: each quadratic term in the energy (such as $\tfrac12mv_x^2$ or $\tfrac12kx^2$) holds on average $\tfrac12kT$. So a monatomic gas has $\tfrac32kT$ per molecule, a diatomic gas $\tfrac52kT$ (two rotations added) and a solid about $3kT$ per atom.

Equipartition explains $C_V = \tfrac32R$ and $\tfrac52R$ from the gas laws, and the root-mean-square speed $\sqrt{3kT/m}$. It fails when $kT$ is small compared with the spacing of quantum levels: vibrations are frozen out at room temperature.

Light reflected from the top and the bottom of a thin transparent film interferes. Two things set the phase difference between the two reflected rays.

• Extra path: at near-normal incidence the lower ray travels an extra $2t$ inside the film, which counts as an optical path of $2nt$ ($n$ is the film's refractive index).
• Phase change on reflection: reflection off a medium of higher refractive index adds a phase change of $\pi$ (half a wavelength). Reflection off a lower index adds nothing.

Count the $\pi$ changes. If exactly one ray has one (soap film in air), bright reflection needs $2nt = (m + \tfrac12)\lambda$ and dark needs $2nt = m\lambda$. If both or neither have one (a coating on glass), the conditions swap.

A wedge of angle $\alpha$ gives straight fringes spaced $\lambda/(2n\alpha)$ apart. A lens on a flat plate gives rings (Newton's rings) whose radii grow as $\sqrt m$. A gap that is very thin compared with $\lambda$ looks dark in reflection when only one reflection has the phase change.

The electric flux through a surface counts field lines crossing it: for a uniform field $E$ at right angles to an area $A$, the flux is $EA$. Gauss's law says the total flux out of any closed surface equals the charge inside divided by $\varepsilon_0$:

It is always true, but it only gives the field when symmetry tells you $E$ is constant over a well-chosen surface.

• Sphere or point charge: outside, $E = \dfrac{Q}{4\pi\varepsilon_0r^2}$ (as if all charge were at the centre). Inside a hollow charged shell, $E = 0$. Inside a uniformly charged solid sphere, $E$ grows in proportion to $r$.
• Long line with charge per length $\lambda$: $E = \dfrac{\lambda}{2\pi\varepsilon_0r}$.
• Large flat sheet with charge per area $\sigma$: $E = \dfrac{\sigma}{2\varepsilon_0}$ on each side, independent of distance.

Conductors in equilibrium: $E = 0$ inside the metal, all excess charge sits on the surface, the whole conductor is at one potential, and just outside the surface $E = \sigma/\varepsilon_0$, perpendicular to it. A charge inside a hollow conductor induces an equal and opposite charge on the inner wall.

Every short piece of wire $\mathrm{d}\vec\ell$ carrying current $I$ makes a field at a point a distance $r$ away given by the Biot-Savart law:

Add (integrate) over the whole circuit. Ampère's law is the shortcut when there is symmetry: round any closed loop, $\oint\vec B\cdot\mathrm{d}\vec\ell = \mu_0I_{\mathrm{enclosed}}$.

Results to know ($\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}$):

• Long straight wire: $B = \dfrac{\mu_0I}{2\pi r}$, circling the wire (right-hand grip).
• Centre of a circular loop of radius $R$: $B = \dfrac{\mu_0I}{2R}$; for $N$ turns multiply by $N$.
• Long solenoid with $n$ turns per metre: $B = \mu_0nI$ inside, almost zero outside.
• Parallel wires a distance $d$ apart: force per metre $\dfrac{\mu_0I_1I_2}{2\pi d}$, attractive for currents in the same direction.

A small current loop of area $A$ acts like a magnetic dipole of moment $m = IA$; in a field $B$ it feels a torque $mB\sin\theta$. Fields from several wires add as vectors, so draw each one's direction before you add magnitudes.

With a sinusoidal supply of angular frequency $\omega$, a capacitor and an inductor each limit the current, like resistors, but they also shift its phase. Their reactances are

In an inductor the current lags the voltage by $90^\circ$; in a capacitor it leads by $90^\circ$. In a resistor they are in phase.

For $R$, $L$ and $C$ in series, add the voltages as rotating arrows (phasors). The result is an impedance

with $I_{\mathrm{rms}} = V_{\mathrm{rms}}/Z$. Complex numbers do the same job: $Z_R = R$, $Z_L = i\omega L$, $Z_C = 1/(i\omega C)$, combined with the ordinary series and parallel rules.

• Resonance: $X_L = X_C$ at $\omega_0 = 1/\sqrt{LC}$, where $Z = R$ and the current is largest.
• Power: only the resistor dissipates energy on average, $P = I_{\mathrm{rms}}^2R = V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi$.

Special relativity rests on two postulates: the laws of physics are the same in every inertial frame, and light travels at $c$ in every inertial frame. Together they force time and space to depend on the observer.

Define the Lorentz factor $\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$. It is 1 at rest, 1.25 at $0.6c$ and grows without limit as $v \to c$.

• Time dilation: a clock moving at $v$ ticks slowly. If it records a proper time $\Delta\tau$ (time between two events at the same place in its own frame), you measure $\Delta t = \gamma\Delta\tau$.
• Length contraction: a rod with proper length $L_0$ (measured at rest) moving along its length has length $L_0/\gamma$ for you. Lengths at right angles to the motion do not change.
• Relativity of simultaneity: events that happen at the same time in one frame, at different places, need not be simultaneous in another.

The commonest mistake is applying the formulas the wrong way round. Always ask: in which frame do the two events happen at the same place? That frame measures the proper time. In which frame is the object at rest? That frame measures the proper length.

Frame S' moves at speed $v$ along the $x$-axis of frame S, with the origins together at $t = t' = 0$. An event at $(x, t)$ in S has coordinates in S'

For differences between two events, replace $x$ and $t$ by $\Delta x$ and $\Delta t$. The quantity $c^2\Delta t^2 - \Delta x^2$ (the interval) is the same in every frame.

• Velocity addition: an object moving at $u$ in S moves at $u' = \dfrac{u - v}{1 - uv/c^2}$ in S'. Equivalently, if it moves at $u'$ in S', then in S it moves at $\dfrac{u' + v}{1 + u'v/c^2}$, which never exceeds $c$.
• Relativistic Doppler effect: for a source approaching at speed $\beta c$, $f = f_0\sqrt{\dfrac{1+\beta}{1-\beta}}$; for a receding source swap the signs. For small $\beta$ this gives the school result $\Delta f/f \approx \beta$.

The $vx/c^2$ term in $t'$ is the source of the relativity of simultaneity: a time difference appears from a space difference alone.