Ranking four circular motions

Problem

Ranking. Four objects follow circular paths with given radius $r$ and speed $v$: (A) $r = 4\;\mathrm{m}$, $v = 2\;\mathrm{m/s}$. (B) $r = 1\;\mathrm{m}$, $v = 3\;\mathrm{m/s}$. (C) $r = 2\;\mathrm{m}$, $v = 2\;\mathrm{m/s}$. (D) $r = 0{,}5\;\mathrm{m}$, $v = 2{,}5\;\mathrm{m/s}$. Order the centripetal accelerations $a = v^2/r$ from smallest to largest.

Solution

Formula. The centripetal acceleration is $a = v^2/r$. Note: $v$ appears squared, so it carries more weight than the radius. Computing for each case:

Case	$r$ (m)	$v$ (m/s)	$a = v^2/r$ (m/s²)
A	$4$	$2$	$4/4 = 1{,}0$
B	$1$	$3$	$9/1 = 9{,}0$
C	$2$	$2$	$4/2 = 2{,}0$
D	$0{,}5$	$2{,}5$	$6{,}25/0{,}5 = 12{,}5$

Ordering (from smallest to largest): $1{,}0 < 2{,}0 < 9{,}0 < 12{,}5$, i.e.

$\ev{A < C < B < D}$

Remark. D wins despite having an intermediate speed, because it has the smallest radius: cornering tightly at a sustained speed requires a lot of centripetal acceleration. A loses despite its non-small $v$, because the large radius “dilutes” the curvature.

Links

Topics: Dynamics Concepts: Centripetal acceleration · Uniform circular motion Skills: Ranking reasoning · Symbolic set-up