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)

    “Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
    —Mary Barnett Gilson (1877–?)