Homework SoursePROBLEM Mr and Mrs Gale end their day at the carnival with a
Mr. and Mrs. Gale end their day at the carnival with a ride on the Ferris wheel. They board their passenger car from a platform and sit on a bench that is 5 ft. above the ground and 5 ft. below the roof of the passenger car, which is 25 ft. from the center of the Ferris wheel. Let (x,y) be the position of the couple at t seconds. The x-axis is the ground and the y -axis is the line perpendicular to the ground through the center of the wheel. When the last passenger\'s have boarded and the ride begins, Mr. and Mrs. Gale are 15 ft. above the ground and 20 ft. to the right of the y-axis, as shown in the picture (i.e., (x,y) = (20,15) at t = 0). The Ferris wheel turns counter-clockwise and makes one revolution every 2 minutes.
Assuming the Ferris wheel continues at a constant speed, write a pair of parametric equations for x and y in terms of t.
Additional Info: In case this is hard for you to picture, the center of the Ferris wheel is 35 ft. up on the y-axis (0,35) with a radius of 25. The starting point for the Gale couple is (20,15) and because their cabin is always straight up on the wheel at any point, their motion is different than that of the wheel. I need to find out the parametric equation for their path. Thanks!
Solution
This is an example of harmonic motion. The general equation for position vs time is shown below.
Position = A * cos ( * t) or Position = A * sin ( * t)
The cos or sin depends on the initial position of the object.
A is the amplitude, which is the maximum distance that the object moves in one direction or the opposite direction. As the Ferris wheel rotates the passengers move around a 25 m radius circle. So, the amplitude is 25.
= angular velocity
The angular velocity is in radians per second. The Ferris wheel turns counter-clockwise and makes one revolution every 2 minutes.
1 revolution = 2 radians ,2 minutes = 120 seconds,Angular velocity = 2 ÷ 120 = /60 radians/second
t = time
x -position = 25 * cos (/60 * t) or x- position = 25 * sin (/60 * t)
x -position = 25 * cos (/60 * t) + 35 or y-position = 25 * sin (/60 * t) + 35
