Advanced Functions Help

Question

the ferris wheel at an amusement park measures 16m in diameter. the wheel does 3 rotations every minute. the bottom of the wheel is 1m above the ground... a) determine the simplest equation that models Megan's height above that ground(h) over time (t). give 2 more equations that model the situation. please explain! :) is the period 0.3 or 20 mins? shouldnt it 0.3 because (1 revolution per a min

Answer 1

First, 3 rotations per minute is a frequency and so 1 minute per 3 rotations or 0.333 minutes/rotation is the period or if you convert to seconds (0.333 * 60) = 20 seconds is the period of rotation.

 

Now, the wave equation for an oscillation like this will be:

 

h(t) = A sin (ωt + φ) + D where A is the amplitude of the wave, ω is the angular frequency which equals 2π * f where f is the frequency (inverse period), t is your time variable, φ is the phase shift, and D is the offset from the axis

 

Since there isn't any initial condition given, there's no need to have a phase shift, so set φ=0

Since the period is 20 seconds, the frequency is 1/20 = 0.05 s^-1

 

Therefore ω will become 2π*(0.05) = 0.1π

The amplitude (A) is 8. You get this because it is half the diameter of the ferris wheel. So at the lowest point, Megan will be 8 meters below the center and at the highest point, Megan will be 8 meters above the center.

 

Finally, the offset from the axis. Since the bottom of the ferris wheel is 1 meter off the ground, we can say that the center of the ferris wheel must be 1+radius = 1+8 = 9 meters off the ground, and thus D = 9

 

So putting that all together:

 

h(t) = 8 sin (0.1π*t) + 9

 

Now, you also wanted two more equations.

First, you could use the cosine function. Since it matters not when you start h(t) because no initial conditions are given, you could just as easily say:

 

h(t) = 8 cos (0.1π*t) + 9

 

Now remember what we said about phase shifts?

 

Well, all a phase shift is is moving the same graph to the right or left along the X-axis, so you could say:

 

h(t) = 8 sin (0.1π*t + π) + 9

 

and now you have the original equation shifted π radians. 

 

Note that if you shift it π/2 radians, you'll have created the cosine version... because cosine is π/2 shifted from sine! Cool, huh?

 

Hope this helped!