(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For my math class, I had to come up with a real-life scenario where data that is extracted from the scenario can produce a sinusoidal function, and I have to graph the function. My scenario involves a person bouncing up and down on a trampoline, while always staying in contact with it. At rest, the taut fabric that stretches over the steel frame of the trampoline is horizontal; this horizontal is the reference level. My graph features elapsed timetas the independent variable, and the height of the person above the reference levelhas the dependent variable. When I presented my idea to my teacher, he told me that the graph could not be a sinusoidal function, and I argued otherwise—that it is possible for the person to simulate simple harmonic motion on the trampolineas long as the person always stays in contact with the trampoline so that gravity is never the only force acting on the person as they bounce up; the only problem was that my scenario was incomplete—I must figure out the minimum value ofhwhen the person dips below the reference level so that the graph is sinusoudal, i.e., so that the motion of the person on the trampoline is simple harmonic motion; I need to do this to determine the amplitude and equilibrium position of the sine wave so that I can come up with a reasonable equation. I'm currently in grade 12 and I took grade 12 physics in grade 11, so I forget some of the stuff, but I invented the following values: say the person, of mass m = 50 kg, rises 0.40 m above the reference level with each bounce (maximum value ofh= 0.40); say the trampoline has 20 springs, each of force constantk= 10 N/m; and say that they-component ofx= 0.03 m. What would be the minimum value ofhwith each bounce? I'm pretty sure that the velocities are irrelevant, as we're dealing with the minimum and maximum positions of the person, wherevandv` are both zero. So,

m= 50 kg

k= 10 N/m (there are 20 springs)

xy = 0.03 m

h= 0.40 m

g= 9.8 m/s²

x` = ?

h` = ?

I'm pretty sure that x` cannot equal x, since at the maximum gravity is acting with the force applied by the springs, and at the minimum gravity is acting against the force applied by the springs. I could use a simpler scenario, but I hate losing arguments. Any useful input would be greatly appreciated.

2. Relevant equations

At the maximum ofh,mg= –20kx; at the minimum,mg= 20kx` (I may be wrong here)

A Conservation of Energy equation(?)

3. The attempt at a solution

Solve forxy`:

(net)Fy = (net)Fy`

mg+ 20kxy =mg– 20kxy`

xy` = –xy = –0.03 m

Then solve forh`:

E=E`

mgh+ 10kxy² = –mgh` + 10kxy`²

–h` = [mgh+ 10k(xy² –xy`²)]/mg

=mgh/mg

–h` =h= 0.40 m

I geth= –h`

-Paul

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# Trampoline Problem!

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