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Question: 1 ptf a string wound around a fixed circle is...

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(1 pt)f 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 Assume the circle is the unit circle 2 +y1 and that the tracing point P, that is the end of the string, starts at the point (1,0) As the string is unwound, the first time it just touches the circle at (1/(2),1/(2)) the end P P(t is now at the point given by (1/V2,1/v2+(1/V2-1/v2) Here the first summand gets to the point of tangency where the string just touches the circle. The scalar multiple π/ 4 is the arc length of the circle over which the string has been unwound. The vector it scales is the (unit vector) direction in which the string points from that point of tangency. Next when he pol t of tangency s at 0. the point along the taut string is direct y right of that point at a distance of π since that s the arc ength hat has been unwound Draw the unit circle and draw an angle t € (0. π /2) at the origin O intersecting the unit circle at a point Q(t- cos t sin t Draw QP Q(t)P(t) tangent to the cicle at (t) with length equal to the length of string that has been unwound. The length QP s A unit vector in the direction of QPis and so (Hint. Use the fact that QP is perpendicular to O@ to find the angle between Q P and the horizontal.) Now use to give P- P(t) P1(t) Pe(t) (P(t), P2(t))

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