Finding the equation that gives the height of the ferris wheel as a function of time.

Question

A ferris wheel has a radius of 10 m, and the bottom of the wheel makes passes 1 m above the ground. If the ferris wheel makes one complete revolution ever 20 s, find an equation that gives the height above the ground of a person on the ferris wheel as a function of time.

Answer 1

I drew the situation, where circle represents ferris wheel turning clockwise. Take a look at the following figure:

https://drive.google.com/file/d/0BwfqpqJa4_wibmRETnFvZ2xrUTg/view?usp=sharing&resourcekey=0-j1Nf-G3ttCLnt_VfIPb0KA

 

Let's say the height of a person on ferris wheel at time t is h (shown by point A)

To calculate h, we need to find out length CD

If angle ACD = θ (which is a function of time, t)

then, CD = 10Cosθ

therefore, h = length OD = OC - CD = 11-10Cosθ

h = 11-10Cosθ   eq1

 

Now, we need to find out how θ is changing as a function of time, t

Since, ferris wheel makes a complete revolution (360 deg turn) in 20 sec

This means θ(t) = (360/20)*t = 18t in degrees

Let's write the angle in radians: since 180 deg = PI radians => 1 deg = PI/180 radians

therefore, θ(t) = 18t*PI/180 = PI*t/10 radians

Plug this value in eq1

h(t) = 11-10Cos(PI*t/10)

 

 

Answer 2

Things we know:

 

radius: r = 10 m

distance between ground and bottom of wheel: d = 1 m

period: T = 20 (s/rev)

 

First we should find the angular frequency. This is usually listed as ω, the lower case Greek letter omega, and represents the number of radians per second. To find this we need to know the period and the number of radians in a complete revolution.

 

Since there are 2π radians  in every revolution we can show:

 

1. ω = (2π radians / 1 revolution)*(1 revolution / 20 seconds) = (π/10) radians per second

 

Let's think about this. Now that we know the angular frequency. We have a number that gives us radians when multiplied by time (must be in seconds). By using a sin or cos function we can input the number of radians and return the position on a unit circle.

 

So far our equation is:

 

    2. y = cos([π/10]t)

 

At t=0 s, y=1. Then at t = 5 s (one quarter turn of the unit circle) y = 0. We then try t = 15 s which yields y=-1, and t=20 s brings us back to y=1. Note that with cos or sin any values will fall between 0 and 1. In order to get our ferris wheel we need to multiply by 10.

 

Now our ferris wheel equations is:

 

    3. h = 10cos([π/10]t)

 

We are not just multiplying by 10. Remember that we are multiplying by 10 meters. Therefore this equation now returns height, denoted h. Let's figure out our height at each time.

 

t=0 s ---> h=10 m

t=5 s ---> h=0 m

t=10 s ---> h=-10m

t=15 s ---> h=0 m

 

Then the pattern repeats. We do not need this, but notice that this equation does not tell us anything about the direction of rotation.

 

So far we have been assuming that we have a circle with a radius of 10 m centered on our coordinate axis. This explains why we are getting positive and negative heights. Essentially we have planted our ferris wheel in the ground. All we need to do is add 11 meters to the equation to fix this. This will "shift" the unit circle up by 11 meters. It must be 11 meters because the center of the ferris wheel is 11 meters above the ground (the radius plus the 1 meter distance to the ground).

 

The final equation becomes:

 

h = 10cos([π/10]t)+11

 

The last thing that we should cover is the phase shift. Notice that at t=0 this equation puts you at 21 meters above the ground. This is exactly the top of the ferris wheel. I don't know many ferris wheels that load from the top. All we need to do is add π radians to the part of our equation inside of cos.

 

A more common equation for a ferris wheel may be:

 

h = 10cos([π/10]t + π) + 11

 

If t=0 we are now taking the cos(π). Our ferris wheel functions like it should.

 

t=0 s ---> h=1 m

t=5 s ---> h=11 m

t=10 s ---> h=21 m

t=15 s ---> h=11 m

t=20 s ---> h=1 m

 

Hope this helps! Good luck!