# Equation for Pendulum Ride

• Jun 16th 2011, 05:33 AM
starshine84
Equation for Pendulum Ride
At Canada's Wonderland, a thrill seeker can ride the Xtreme
Skyflyer. This is essentially a large pendulum of which the rider is the bob. The height of the rider is given for various times:

Time(s) 0 1 2 3 4 5 6 7 8 9
Height(m) 55 53 46 36 25 14 7 5 8 15

a. Find the amplitude, period, vertical translation, and phase shift for this function. [Note: the table does not follow the bob through one complete cycle, so some thought will be required to answer this question.]

b. Determine the equation of the function in the form: h(t) = asin[b(t - c)] + d.

From the table of values I know that the max value is 55m because that is the starting point of the ride and when the rider is going to be at his or her highest point. I know that the lowest point is 5m at 7 seconds. After that point, the rider starts to swing back up to complete the second half of the cycle.

amplitude = max-min/2
= 55-5/2
= 50/2
= 25

If it takes 7 seconds for the rider to get from the highest point to the lowest point, then then entire cycle will take 14 seconds to complete. Therefore, the period will be 14.

period = 360/b
14 = 360/b
b = 360/14
b = 25.71

d = max+min/2
= 55+5/2
= 60/2
= 30

I think I'm on track so far, but I'm having difficulty finding the phase shift. I thought I could create a formula based on what I've already found, plug in a coordinate from the table of values for the values of x and y, and then solve for c. I've done this below:

y = a sin (bx-c) + d
y = 25 sin (25.71 x - c) + 30

(1,53)- Let x be 1 and let y be 53 in y = 25 sin (25.71 x - c) + 30

y = 25 sin (25.71 x - c) + 30
53 = 25 sin (25.71 (1) - c) +30
23 = 25 sin (25.71 (1) - c)
0.92 = sin (25.71 - c)
66.92608193 = 25.71 - c
41.22 = -c
-41.22 = c

Therefore, the phase shift would be -41.22, which would move the graph 41.22 degrees to the right. However, when I graph y = 25 sin (25.71 x - 41.22) + 30 the shift is not correct because the highest point in the graph is not at x=0.

Is there another way to calculate phase shift given a table of values? Some simplified formula that I'm just missing?

Thanks!
• Jun 16th 2011, 04:49 PM
skeeter
Re: Equation for Pendulum Ride
I took your given data and computed a regression equation on my TI-84 ...

http://img543.imageshack.us/img543/6195/pendgraf.jpg

http://img405.imageshack.us/img405/4251/pendeq.jpg

assuming 55 is the max height, 5 is the min height, period is 14 seconds, and you just want to get close, I recommend the following equation ...

$\displaystyle h = 25\cos\left(\frac{\pi}{7} \cdot t\right) + 30$

here is this equation graphed against the data ...

http://img845.imageshack.us/img845/7281/pendgraf2.jpg