A gear modeled by a circle of radius 2r is centered at the origin. A second gear modeled by a circle C' of radius r is lying tangent to the first with center (0, 3r). A point P is chosen on the radius between points (0, 3r) and (0, 4r), whose distance from the center of C' is b, where 0 sbsr. Now, the gear C' rolls smoothly in a counterclockwise motion along C, so that the line that Plies on is rotated around the inside of C'. Thus, P traces out a plane curve. A) Given a positioning of the circle C_ let @ be the angle between the line from the center of C' to the origin and the positive x-axis. Write parametric equations the center of the circle c'. B) For the angle and position of C' as above, write a Cartesian equation for the circle of radius b with the same center as that of C'. C) Now, for the angle 0, we would like to find the angle of rotation of the line that P is on with respect to the circle C. Let this angle be a. Find the arclength of the sector of the circle C traced out by 8. Equate this to the arclength of the sector traced out by a. Now, adding a to the angle to get the total angle of rotation, conclude that the total angle of rotation of Lin total is 30. D) Conclude that the parametric equations for P are given by x = b cos(30) + 3r cos(), y = b cos(38) + 3r cos(). A gear modeled by a circle of radius 2r is centered at the origin. A second gear modeled by a circle C' of radius r is lying tangent to the first with center (0, 3r). A point P is chosen on the radius between points (0, 3r) and (0, 4r), whose distance from the center of C' is b, where 0 sbsr. Now, the gear C' rolls smoothly in a counterclockwise motion along C, so that the line that Plies on is rotated around the inside of C'. Thus, P traces out a plane curve. A) Given a positioning of the circle C_ let @ be the angle between the line from the center of C' to the origin and the positive x-axis. Write parametric equations the center of the circle c'. B) For the angle and position of C' as above, write a Cartesian equation for the circle of radius b with the same center as that of C'. C) Now, for the angle 0, we would like to find the angle of rotation of the line that P is on with respect to the circle C. Let this angle be a. Find the arclength of the sector of the circle C traced out by 8. Equate this to the arclength of the sector traced out by a. Now, adding a to the angle to get the total angle of rotation, conclude that the total angle of rotation of Lin total is 30. D) Conclude that the parametric equations for P are given by x = b cos(30) + 3r cos(), y = b cos(38) + 3r cos()
