Mr. Ferris' Rides

In analyzing this problem, our first task is to create a graph of the movement of the Ferris wheel. We know from physics that the standard equation for this type of motion, called SHM or Simple Harmonic Motion, is given by

s = a sin(2pft + f) + s₀

where

• s = displacement
• a = amplitude
• f = frequency
• t = time
• f = rotation shift
• s₀ = height of the center of the circle

Note: The calculator must be in radians mode for this exercise. See Fig I.1 below.

Fig I.1

Letting s be the y-axis variable and t be the x-axis variable, our particular Ferris wheel will have the following specifications:

• a = 10 meters
• f = ¹/₁₂₀ rev/sec
• f = - ^p /₂
• s₀ = 11 meters

We are using f = - ^p /₂ because the passengers will be embarking at the bottom of the wheel, rather than the right edge.

Another way of writing f = - ^p /₂ is to replace the function sine with the function -cosine. Using this function, the equation for our Ferris wheel would become

y = -10 cos(^px/₆₀) + 11

Now, enter the equation into the Y= menu. Remember that the

and

keys perform different functions. Be sure that you use the correct one. Your window should look like Fig I.2 shown below. If you need help in graphing functions, see the lesson on graphing with the TI calculator.

Fig I.2

You should now change your window settings to match the parameters of the Ferris wheel.  Xmin = 0, Xmax = 240, Ymin = 0, Ymax = 25. Your window should look like Fig I.3 shown below.

Fig I.3

Now graph the equation.

One day, Mr. Ferris decided to upgrade his Ferris wheel, but forgot to give you the updated specs. To determine the changes, you decide to ride the wheel and take some measurements which are listed in the following table. (Fig II.1)

time	height
0 s	2 m
45 s	13.5 m
90 s	25 m
135 s	13.5 m
180 s	2 m

Fig II.1

Enter the data into the calculator lists. Entering data into lists is discussed in the graphing calculator lesson called Data Input. Please refer to it if you have trouble with this section). After the data is entered (Fig II.2),

Fig II.2

press , select the last item on the list, C: SinReg (Fig II.3), and press enter twice.

Fig II.3

This will produce a calculator estimate of the equation for the wheel (Fig II.4).

Fig II.4

Now enter the equation into the Y= menu, in the Y₂ field, via the Vars menu (Fig II.5).

Fig II.5

Note, this is only an ESTIMATE of the equation. The real equation is much more concise. If you look at the graph, you'll see the new equation being graphed over top of the old one, as in Fig II.6.

Fig II.6

As you can see, the period of rotation is longer, the amplitude is higher, and the starting position is higher.

Mr. Ferris has once again decided to change his wheel. Instead of having his passengers embark at the bottom of the wheel, they will now get on after the wheel has completed an 8th of a turn with respect to the original lowest position - the bottom of the circle.

Since - ^p /₂ equals one 4th of a turn, an 8th of a turn would be represented by - ^p /₄. The new equation for the wheel's behavior could be written as either

y = - 11.5 cos(^px/₉₀ + ^p /₄) + 13.5

or

y = 11.5 sin(^px/₉₀ - ^p /₄) + 13.5

Either equation will produce the same result, but for the sake of simplicity, enter the second equation into the Y= menu under Y₃ (Fig III.1)

Fig III.1

Now graph it. The result should look like Fig. III.2.

Fig III.2

The equations you have been looking at so far are only for the vertical motion of the Ferris wheel, however, there is also horizontal motion to be taken into account. This is actually much more simple than it may initially seem. All that is required is a 4th of a turn of the wheel. When embarking on the Ferris wheel from the same point, you would already be a little past the midpoint of the horizontal movement, and increasing toward the upper extreme of that cycle. An easy way to graph this is to simply replace sine with cosine in the equation.

Clear the top two equations from the Y= menu, and enter the following equation into the Y₁ field:

y = 11.5 cos(^px/₉₀ - ^p /₄) + 13.5

Of course we must remember that in our current interpretation of this event, the y in Y₁ actually means the horizontal motion of the wheel as measured from the center of the wheel. See Fig IV.1 shown below.

Fig IV.1

When graphed, it should look like Fig IV.2.

Fig IV.2

Now that you have the graphs for both vertical and horizontal motion, you can find a passenger's instantaneous velocity in either the horizontal or vertical dimension by using the same technique as in "Police Chasing Neon." You can then calculate a passenger's instantaneous speed by using the Pythagorean Theorem with the magnitudes of these horizontal and vertical velocities each serving as one leg of the right triangle. The magnitude of the resultant, or absolute velocity, is the length of triangle's hypotenuse. Fig IV.3 shows this done at x = 120 seconds.

Fig IV.3

Thus far, you have been analyzing each dimension of the movement of the Ferris wheel separately. The calculator includes a mode with which to analyze two dimensional motion. This is called "Parametric Graphing." To put your calculator into Parametric mode, press , and select "Par" (Fig V.1).

Fig V.1

Now go back to the Y= menu. You will see that is has changed. Instead of Y₁, Y₂, Y₃, there is now an X_#T and a Y_#T for each entry. Put the equation for the horizontal motion of the Ferris wheel into the X_1T field, and the equation for the vertical motion of the Ferris wheel into the Y_1T field. Note that the x in your original y = equations should be replaced with a T whenever using parametric mode. See Fig V.2 shown below.

Fig V.2

You will have to change your window as well. A good window for this graph is Tmin = 0, Tmax = 180, Tstep = 1, Xmin = 0, Xmax = 25, Ymin = 0, Ymax = 25 (Fig V.3).

Fig V.3

Now look at the graph.