. Preliminaries to the Rolling Wheel Problem We will now apply the material of this chapter to the problem of finding the x and y coordinates of a specified point on a wheel after it has rolled a given distance. After a introductory illustrative example, the other examples in this section will require use of the trigonometric identities for sin(a + b) and cos(a + b) which.we will introduce. Illustrative Example 1: A wheel of radius 2" rolls 15x" to the left. If the initial coordinates of a point on the wheel relative to the center of the wheel are P(2, 0). (The center is the origin of a Cartesian coordinate system.). Find the final coordinates of this point relative to the center of the wheel after it rolls 15m" to the left. 2 P(2,0) final position (?) initial position S = 151" Fig. 1 Solution: In order to find the final position of P we first determine the angle 0 through which the spoke Of rotates as the wheel rolls 151" Using the formula s = 10 for the distance s the center of a rolling wheel moves when a spoke on this wheel rotates through an angle of O radians, we have: s = re 151 = 20 0 = = 15n - rad 2 Because the wheel is rolling counterclockwise, O is a positive angle. Also, the default units for O is radians when using this formula. To locate the position of the spoke OP after it has rotated 0 = - rad, we rewrite 2" = (7 + _) n = 7n+"In the exercises to follow we are given the coordinates of a point P on a wheel in its initial position. We are also given the radius of the wheel and the distance s that it rolls to the left. In each case, use the method illustrated in the text to find the final coordinates of P, relative to the center of the wheel. radius of wheel initial coordinates of P distance wheel rolls 1. 5 (3,4) (15m) 2. 4 (2,2V 3) (15m)" 3. 4 (3, V7) (13)"