Trampoline Problem: Find Minimum h Value for Sinusoidal Function

[Jalhalla]

Homework Statement

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 time t as the independent variable, and the height of the person above the reference level h as 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 trampoline as 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 of h when 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 of h = 0.40); say the trampoline has 20 springs, each of force constant k = 10 N/m; and say that the y-component of x = 0.03 m. What would be the minimum value of h with each bounce? I'm pretty sure that the velocities are irrelevant, as we're dealing with the minimum and maximum positions of the person, where v and v` 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.

Homework Equations

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

A Conservation of Energy equation(?)

The Attempt at a Solution

Solve for xy`:

(net)Fy = (net)Fy`
mg + 20kxy = mg – 20kxy`
xy` = –xy = –0.03 m

Then solve for h`:

E = E`
mgh + 10kxy² = –mgh` + 10kxy`²
–h` = [mgh + 10k(xy² – xy`²)]/mg
= mgh/mg
–h` = h = 0.40 m

I get h = –h`

-Paul

 

[rcgldr]

The surface of the trampoline also flexes. Resting point isn't "horizontal", but the point below horizontal when the person is just standing and not bouncing. To make this easier, consider the trampoline to be a single spring that the person is standing on.

 

[Jalhalla]

When I say "at rest", I mean when the person is not standing on it. I've edited my original post to show my work. But if the reference level is 40 cm below the horizontal, and the person bounces 40 cm up to reach maximum h, then xy` = 0? *Sigh* ... Or does the person simply bounce 40 cm up and then return to his starting position after coming back down, and the equilibrium is –20? Even so, xy` <> 0 ... I need people to show me some equations, please.