ferris wheel problem?

Question

a ferris wheel is 40 meters in diamter and boarded at ground level. the wheel completes one full revolution every 8 minutes. At t=0 you are in the three o'clock position and descending. Write an equation for the curve.

Answer 1

If x = 0 is ground level, x increasing to the right (towards three o'clock), and y = 0 where boarding, increasing upwards, both x and y in meters, then the equation of the wheel is x^2 + (y - 20)^2 = 20^2.

At t = 0 one is three o'clock or zero radians, descending means clockwise or radians decreasing, 2 pi radians taking 8 minutes.

So with t in minutes, x(t) = 20 cos (- 2 pi t / 8), y(t) = 20 (1 + sin(- 2 pi t / 8)).

Dan

Answer 2

Circular motion is a form of simple harmonic motion, which can be modeled by either y = AsinBx or y = AcosBx, depending on whether the moving object begins at the "middle" height or an extreme height. Since the car of the Ferris wheel is beginning at the 3 o'clock position, y=AsinBx can be used.

A is the amplitude of the motion. In this case that means that A half of the distance from the top to the bottom of the Ferris wheel. Therefore, A = 20 meters.

B deals with the period of the function. If you call the period T, you say that T = (2*pi)/B. Since the wheel completes one cycle in 8 minutes, we can say that B = 8 (the period of a simple harmonic motion is the time necessary to complete one full cycle). Therefore, T = (2*pi)/8 = pi/4

So, we say that the height of the car is modeled by y = 20sin((pi/4)t).

Except for two things.

First, the car is descending, meaning the function is decreasing from t = 0. Since the untransformed sine function is increasing from t = 0 to t = pi/2, we use A = -20 instead.

Second, the initial height is 20 meters. The extreme heights of the wheel are 0 meters and 40 meters. Since sin(0) = 0, we need an upward adjustment of 20 meters.

So, the height of the car can be modeled by the function,
y = 20 - 20sin((pi/4)*t)