Complete translation–rotation analogy

One of the most beautiful things in mechanics is that the laws of rotation have exactly the same structure as those of translation. It is not an approximate resemblance: it is a perfect translation dictionary, where every translational quantity has a rotational counterpart that inherits all its properties.

The translation ↔ rotation dictionary

Translation	Rotation	Relation
Position $s$	Angle $\theta$	$s = r\theta$
Velocity $v$	Angular velocity $\omega$	$v = r\omega$
Acceleration $a$	Angular acceleration $\alpha$	$a_t = r\alpha$
Mass $m$	Moment of inertia $I$	$I = \sum m_i r_i^2$
Force $\vec{F}$	Moment $M$	$M = Fr\sin\theta$
$\sum F = ma$	$\sum M = I\alpha$
$\vec{p} = m\vec{v}$	$L = I\omega$
$E_\text{cin} = \tfrac{1}{2}mv^2$	$E_\text{cin,rot} = \tfrac{1}{2}I\omega^2$
Impulse $\vec{I}$	Angular impulse $I_\text{ang}$

The equations of uniformly accelerated rotational motion

Since $\theta$, $\omega$, $\alpha$ stand to $s$, $v$, $a$ as their twins, the equations of motion for uniformly accelerated rotation with $\alpha$ constant are the same as for uniformly accelerated straight-line motion, word for word:

$\theta_B - \theta_A = \omega_A(t_B-t_A) + \tfrac{1}{2}\alpha(t_B-t_A)^2$

$\omega_B - \omega_A = \alpha(t_B - t_A)$

$\omega_B^2 - \omega_A^2 = 2\alpha(\theta_B - \theta_A)$

There is nothing new to learn: you just swap the symbols. Whoever has understood straight-line kinematics has already understood, for free, rotational kinematics.

An analogy that never ends

The analogy is so deep that every theorem of translational mechanics has a rotational counterpart: conservation of momentum $\vec{p}$ becomes conservation of angular momentum $L$, the impulse theorem becomes the angular impulse theorem, and even the Atwood machine has a rotational analogue with massive pulleys. Every time you meet a formula for a rotating body, try to recognise which translational formula is “redoing its make-up” in angular coordinates.

Collegamenti

Argomenti: Dinamica rotazionale Concetti: Momento angolare · Energia cinetica rotazionale Metodi: Analogia traslazione-rotazione

Esercizi collegati: Esercizio svolto — sfera che rotola giù da un piano · Problema — Momento torcente rettangolare e L(t) · Problema — Asta incernierata che cade dall’orizzontale