Arc Length with Parametric Equations — Question 9

Question 9

Problem

A particle goes around the unit circle twice. Is its path length $2\pi$ or $4\pi$? Justify using $x=\cos2t,y=\sin2t$, $0\le t\le2\pi$.

Question 9 – Solution

Solution

1. Differentiate: $$x'(t)=-2\sin2t, \qquad y'(t)=2\cos2t.$$

2. The speed is constant: $$\sqrt{x'^2+y'^2} =\sqrt{4\sin^22t+4\cos^22t} =2.$$

3. Integrate over the full parameter interval: $$L=\int_0^{2\pi}2\,dt =\boxed{4\pi}.$$

4. The angle of the point is $2t$, which increases from $0$ to $4\pi$. Therefore, the unit circle is traversed twice. Its geometric circumference is $2\pi$, but the particle travels that circumference twice, producing total traveled length $4\pi$.