What is the Minimum Height for Simple Harmonic Motion on a Trampoline?

[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'm 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

 

[catkin]

Your teacher is right because the trampoline is not a linear spring -- it gets stiffer with extension as the springs move nearer to the vertical.

 

[Jalhalla]

Then let's say the springs are vertical.

 

[catkin]

If the effective stiffness of the (redesigned!) trampoline is a constant it will work (and is no different from the classic mass-on-a-spring explained in so many places).

If h is not equal to h' then the motion is not sinusoidal.

 

[Jalhalla]

If the effective stiffness of the (redesigned!) trampoline is a constant it will work (and is no different from the classic mass-on-a-spring explained in so many places).

If h is not equal to h' then the motion is not sinusoidal.

I don't see how |h| = |h´|, because, at h, the force of gravity acts with the force exerted by the springs, while, at h´, the force of gravity acts against the force exerted by the springs.

 

[Vidatu]

Hate to say it, but this doesn't seem like a real world example anymore. First, you are assuming no damping occurs, which isn't the case. Then, you change the springs' orientation, to something pretty unlikely.

I'd say to redesign your system, as to get thus to work, you'll probably have to make it pretty outlandish. Something like the horizontal displacement of a clock's pendulum, or the change in sunrise/sunset times over a year may be better.