Ranking points on a rotating disc

Problem

Three points on a disc rotating with constant angular velocity $\omega$:

• A at distance $r$ from the centre;
• B at distance $2r$;
• C at distance $3r$.

Rank the centripetal accelerations from smallest to largest.

Solution

The right formula. On a rotating rigid body, all points have the same angular velocity $\omega$, but different (tangential) speeds: $v = \omega\,d$, larger the further the point is from the centre. It is therefore convenient to express the centripetal acceleration as a function of $\omega$, which is common to all: $a_c = \omega^2\,d$ ($d$ = distance from the centre). At equal $\omega$, the acceleration is proportional to the distance from the centre.

Calculation for the three points.

Point	Distance $d$	$a_c = \omega^2 d$
A	$r$	$\omega^2 r$
B	$2r$	$2\,\omega^2 r$
C	$3r$	$3\,\omega^2 r$

Ranking (from smallest to largest): $\omega^2 r < 2\omega^2 r < 3\omega^2 r$, i.e. $a_A < a_B < a_C$, that is

$\ev{a_C > a_B > a_A}$

Beware the trap. If one mistakenly used $a_c = v^2/d$ assuming the same $v$ for all, the opposite order would result. But on a rigid disc it is $\omega$ that is common, not $v$: the outermost point (C) has both the greater speed and the greater centripetal acceleration.

Links

Topics: Dinamica Concepts: Accelerazione centripeta · Velocità angolare · Moto circolare uniforme Skills: Ragionamento di ranking · Impostazione simbolica Objects: Disco rotante