Gear Tooth Problem: The surfaces of gear teeth are made in the shape of an involute of a

Chapter 13, Problem C2

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Gear Tooth Problem: The surfaces of gear teeth are made in the shape of an involute of a circle (Figure 13-6e). This form is used because it allows the motion of one gear to be transmitted uniformly to the motion of another. An involute is the path traced by the end of a string as it is unwound from around a circle. In this problem you will see how the polar coordinates of a point on an involute are related to the Cartesian coordinates that can be found by parametric equations. You will do this by using vectors in the form of complex numbers. Figure 13-6e a. Suppose a gear is to have a radius of 10 cm (to the inside of the teeth). The gear tooth surface is to have the shape of the involute formed by unwrapping a string from this circle. Vector (Figure 13-6f) goes from the center of the circle to the point of tangency of the string. Write as a complex number in polar form, in terms of the angle t radians. b. Vector goes along the string from the point of tangency to the point on the involute. Explain why its length is the same as the arc of the circle subtended by angle t. Write as a complex number in polar form. Observe that, with respect to the horizontal axis, the angle for is less than t because is perpendicular to . Use the appropriate properties to get in terms of functions of t. c. Vector is the position vector to point P(x, y) on the involute. Show that = 10(cos t + t sin t) + 10i(sin t t cos t) d. Show that r (the length of ) is given by e. The gear teeth are to be 2 cm deep, which means that the outer radius of the gear will be 12 cm. If the inside of the tooth shown in Figure 13-6f is at t = 0, what will be the value of t at the outside of the tooth? What will be the value of for this value of t? f. Machinists who make the gear need to know the degree measure of angle in part e. Find this measure in degrees and minutes, to the nearest minute.