QQuestionMathematics
QuestionMathematics
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)\right)^{2}+\left(y^{\prime}(t)\right)^{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.
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Answer
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Step 1:I'll solve this cycloid arc length problem step by step:
Step 2:: Find the derivatives of x(t) and y(t)
x'(t) = R(1 - \cos t)
y'(t) = R \sin t
Step 3:: Square the derivatives
[x'(t)]^{2} = R^{2}(1 - \cos t)^{2}
[y'(t)]^{2} = R^{2} \sin^{2} t
Step 4:: Add the squared derivatives
[x'(t)]^{2} + [y'(t)]^{2} = R^{2}(1 - \cos t)^{2} + R^{2} \sin^{2} t
Step 5:: Use trigonometric identity \sin^{2} t + \cos^{2} t = 1
Rearranging, $$\sin^{2} t = 1 - \cos^{2} t
Step 6:: Simplify the integrand
\sqrt{[x'(t)]^{2} + [y'(t)]^{2}} = R\sqrt{2(1 - \cos t)}
Step 7:: Determine integration limits
One complete revolution occurs from $$t = 0$$ to $$t = 2\pi
Step 8:: Set up the arc length integral
L = \int_{0}^{2\pi} R\sqrt{2(1 - \cos t)} dt
Step 9:: Solve the integral
This integral evaluates to $$8R
Final Answer
The arc length of one complete arch of a cycloid is 8 times the radius of the generating circle.
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