Ranking centripetal accelerations

Problem

Four uniform circular motions:

• (A) $v = 5$ m/s, $r = 1$ m;
• (B) $v = 10$ m/s, $r = 1$ m;
• (C) $v = 5$ m/s, $r = 4$ m;
• (D) $v = 20$ m/s, $r = 8$ m.

Rank the centripetal accelerations $a = v^2/r$ from smallest to largest.

Solution

Formula. The centripetal acceleration of a uniform circular motion is: $a = \frac{v^2}{r}$ Note: it depends on the square of the speed, so $v$ matters much more than $r$.

Case-by-case calculation. $a_A = \frac{5^2}{1} = 25\ \mathrm{m/s}^2$ $a_B = \frac{10^2}{1} = 100\ \mathrm{m/s}^2$ $a_C = \frac{5^2}{4} = \frac{25}{4} = 6{,}25\ \mathrm{m/s}^2$ $a_D = \frac{20^2}{8} = \frac{400}{8} = 50\ \mathrm{m/s}^2$

Increasing order. Comparing $6{,}25 < 25 < 50 < 100$: $\ev{C < A < D < B}$

Note: B has the same radius as A but double the speed, and its acceleration is quadruple ($4\times$). C has the same $v$ as A but a radius $4\times$ larger, so an acceleration $4\times$ smaller. The square of the speed dominates.

Linked items

Topics: Dinamica Concepts: Accelerazione centripeta · Moto circolare uniforme Skills: Ragionamento di ranking