The involute of a circle If a string wound around a fixed

Chapter 12, Problem 19E

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The involute of a circle If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end \(P\) traces an  involute of the circle. In the accompanying figure, the circle in question is the circle \(x^{2}+y^{2}=1\) and the tracing point starts at \((1, 0)\).  The unwound portion of the string is tangent to the \(Q\), and  \(t\) is the radian measure of the angle from the positive  \(x-axis\) to segment  \(OQ\). Derive  the parametric equations

\(x=\cos t+t \sin t, y=\sin t-t \cos t, t>0\)

Of the point \(P(x, y)\) for the involute.

Equation Transcription:

Text Transcription:

P

x^2 + y^2 = 1

(1, 0)

Q

t

x-axis

OQ

x = cos t + t sin t,  y = sin t - t cos t,  t >0

P(x, y)