The arc length $L$ of a curve given parametrically by $(x(t), y(t))$ for $a \leq t \leq b$ is given by the formula:

$
L=\int_{a}^{b} \sqrt{\left(x^{\prime}(t) ight)^{2}+\left(y^{\prime}(t) ight)^{2}} d t
$

A path of a point on the edge of a rolling circle of radius $R$ is a cycloid, given by:

$
\begin{aligned}
& x(t)=R(t-\sin t) \
& y(t)=R(1-\cos t)
\end{aligned}
$

where $t$ is the angle (in radians) the circle has rotated.
Find the length $L$ of one "arch" of this cycloid. That is, find the distance traveled by a small stone stuck in the tread of a tire of radius $R$ during one revolution of the rolling tire.

Question

The arc length $L$ of a curve given parametrically by $(x(t), y(t))$ for $a \leq t \leq b$ is given by the formula:

$
L=\int_{a}^{b} \sqrt{\left(x^{\prime}(t)ight)^{2}+\left(y^{\prime}(t)ight)^{2}} d t
$

A path of a point on the edge of a rolling circle of radius $R$ is a cycloid, given by:

$
\begin{aligned}
& x(t)=R(t-\sin t) \
& y(t)=R(1-\cos t)
\end{aligned}
$

where $t$ is the angle (in radians) the circle has rotated.
Find the length $L$ of one "arch" of this cycloid. That is, find the distance traveled by a small stone stuck in the tread of a tire of radius $R$ during one revolution of the rolling tire.

StudyXY · Accepted Answer

I'll solve this cycloid arc length problem step by step: : Find the derivatives of x(t) and y(t) : Square the derivatives : Add the squared derivatives : Use trigonometric identity $\sin^{2} t + \cos^{2} t = 1$ : Simplify the integrand : Determine integration limits : Set up the arc length integral : Solve the integral The arc length of one complete arch of a cycloid is 8 times the radius of the generating circle.

Q
QuestionAnatomy and Physiology

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I'll solve this cycloid arc length problem step by step:

Step 2
: Find the derivatives of x(t) and y(t)

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