HL Deep Dives — Rigid Bodies & Relativity

Rotational analogues of suvat.

$\theta$ in radians; $2\pi$ rad = one revolution.

Rotational form of $F = ma$; N m.

Resistance to angular acceleration; kg m².

Net torque produces angular acceleration.

kg m² s⁻¹. Conserved when net external torque = 0 (figure-skater effect).

Using degrees where radians are required; forgetting that rolling combines translational + rotational KE.

(1) Laws of physics are the same in every inertial frame. (2) The speed of light c is the same for every observer.

$\gamma \ge 1$; $\gamma \to \infty$ as $v \to c$.

Moving clocks tick slow; $\Delta t_0$ = proper time (measured at rest in the events' frame).

Moving objects are shorter along the direction of motion; $L_0$ = proper length (rest frame).

Rest energy + kinetic; $E_k=(\gamma-1)mc^2$.

Collinear motion. Reduces to Galilean $u' + v$ when $u', v \ll c$. With $u' = c$ gives $u = c$ — postulate 2 baked in.

Receding source ($\beta = v/c$); approaching uses the reciprocal under the root. Reduces to $\Delta\lambda/\lambda \approx v/c$ when $\beta \ll 1$. Use when redshift comes from $v$ comparable to $c$.

$\Delta t_0$ measured where the two events occur at the same place; $L_0$ measured in the frame where the object is at rest. Other observers see longer times and shorter lengths.

$\gamma$: dimensionless; $c \approx 3.0\times10^8$ m s⁻¹; energies often quoted in MeV.

Forgetting that $\gamma\to 1$ at everyday speeds; confusing proper vs observed time/length; using Newtonian KE = ½mv² when v is comparable to c.

Solid sphere, no slip. Rotational KE is $\tfrac{2}{7}$ of the total.

Earth frame uses time dilation ($\tau = \gamma\tau_0$); muon frame uses length contraction ($L = L_0/\gamma$). Both predict the same survival fraction.

Use this whenever the non-relativistic shortcut gives $\beta > 0.3$ (or worse, $\beta > 1$).

Same in every frame. For photons ($m = 0$): $E = pc$.