Involute of a Circle 2: Figure 13-5m shows an involute of a circle. A string is wrapped

Chapter 13, Problem 9

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Involute of a Circle 2: Figure 13-5m shows an involute of a circle. A string is wrapped around a circle of radius 5 units. A pen is tied to the string at the point (5, 0). Then the string is unwound in the counterclockwise direction. The involute is the spiral path followed by the pen as the string unwinds. The curve is interesting because gear teeth made with their surfaces in this shape transmit the rotation smoothly from one gear to the next. Figure 13-5m a. The parameter t is the radian measure of the angle between the positive x-axis and the line from the origin to the point of tangency of the string. Vector goes from the center of the circle to the point of tangency. Vector goes from the point of tangency to point P(x, y) on the involute. Find a vector equation for position vector (not shown) to point P in terms of the parameter t. Note that and are perpendicular because is a radius to the point of tangency. Confirm that your equation is correct by plotting it on your grapher. Use at least three revolutions for the t-range. Construct an involute by wrapping a string around a roll of tape, tying a pencil or pen to the end of the string, and tracing the path as you unwind the string. Does the involute you plotted in part a agree with this actual involute?