Ferris Wheel #1) As a Ferris wheel turns, the distance a rider is above the ground varies sinusoidally with time.

Question

Ferris Wheel #1) As a Ferris wheel turns, the distance a rider is above the ground varies sinusoidal with time. When the Ferris wheel begins turning, your seat location is as shown below. It takes you 3 seconds to reach the top. The highest point on the wheel is 43 feet above the ground. The wheel makes a full circle every 8 seconds and has a diameter of 40 feet. Sketch a graph of your height as a rider as a function of time (you do not need to show the graph here if it is too difficult to put on the computer – just graph it to help you write your equation), and answer the following equations.

 

 

 

a) What is your lowest height above the ground as the wheel turns?


b) What is an equation for your graph?


c) Use this equation to predict your height above the ground at 4.5 seconds

Answer 1

Hi Ali,

 

  I have made a figure to help with this problem, see:

   https://www.wyzant.com/resources/files/466421/equation_of_ferris_wheel_height_vs_time

 

We can answer the questions without the figure:

 

a) Since the highest point is 43 feet and the diameter is 40 feet, the lowest point will be 43-40 = 3 feet above ground.

 

b) The radius of the wheel is 20 feet (=40/2), so that will multiply a sin or cos function.

The center of the wheel is 23 high, so that will be the middle of the sin/cos curve.

It takes 8 seconds per revolution, so we'll want the sin/cos to repeat every 8 seconds.

Lastly it is at the top at 3 seconds.  A cos curve is at the top when its argument is 0, so if we use cos with t-3 that should give what we want.

Putting it all together:

 

  height = 23 + 20*cos(360 * (t-3)/8 )

 

where I've assumed the cos is working in degrees.  If it is in radians then in place of 360 put 2pi.

 

c) we just put 4.5 in for t to get the height at 4.5 s.  I'll leave this to you ;-)

(You can roughly check your answer by looking at the sketch in the figure to see if it makes sense.)

 

Answer 2

Ali,

I hope you've already worked this out, but if not:

 

Draw a sine curve through one complete period. If the wheel takes 8 seconds to make a complete turn, then the complete period of the sine curve takes 8 seconds. i.e. from  0 to 1 takes 2 seconds, from 1 back to 0 takes another 2 seconds, from 0 down to -1 takes 2 seconds, and the final leg from -1 up to 0 is 2 seconds.

 

Now since I can't see the picture of where your wheel is starting, I'll have to determine the phase shift from the time to reach the top. But at it's core, this is just a sine function shifted up so that as y is the distance above the ground, we don't get any riders crashing into the ground. We also have the phase shift to account for the rider not starting at the bottom.

 

a) if the highest point is 43' and the diameter is 40', then the lowest point is:

 

43 - 40 = 3' above the ground.

 

 

b) the basic sine function is just y = Asin(x - B) + C, where A is the amplitude, B is the phase shift, and C is the vertical shift. We'll use t instead of x since we're relating time instead of angular displacement. But we also have to relate it to the angular displacement.  For the vertical shift, we need to know how high the center is above the ground. Average of the high and the low is:

 

C = (43 + 3)/2 = 23

 

We know the amplitude is the maximum distance from the center, or radius, so:

 

A = 20

 

Now for the phase shift. Since the total rotation takes 8 seconds, the rider is 3/8 of a rotation from the top. Now the normal sine curve starts at the origin and goes up. That would be the midpoint elevation on the way up, 2 seconds from the top. Another 1 second before that puts the rider at 45º below that, which equates to a phase shift of π/4.

So if B = π/4 then our equation becomes:

 

Y = 20sin(t(2π/8) - π/4) + 23

 

c)  Y = 20sin(4.5(2π/8) - π/4) + 23 = 30.65'

 

Remember, even though it may seem like the phase shift should be negative, moving backward from the normal start, the graph is actually moving to the right which is positive. Hope this helps.