Cycloid - Arc Length

Arc Length

The arc length S of one arch is given by

\begin{align} S &= \int_{t=0}^{t=2 \pi} \left^\frac{1}{2} dt \\ &= \int_{t=0}^{t=2 \pi} r \sqrt{2 - 2\cos(t)} dt \\ &= \int_{t=0}^{t=2 \pi} 2r \sin\left(\frac{t}{2}\right) dt \\ &= 8 r \end{align}

Read more about this topic:  Cycloid

Famous quotes containing the words arc and/or length:

    “You say that you are my judge; I do not know if you are; but take good heed not to judge me ill, because you would put yourself in great peril.”
    —Joan Of Arc (c.1412–1431)

    “Twenty-four-hour room service generally refers to the length of time that it takes for the club sandwich to arrive. This is indeed disheartening, particularly when you’ve ordered scrambled eggs.”
    —Fran Lebowitz (b. 1950)