Trampoline Problem: Find Minimum h Value for Sinusoidal Function

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In summary, the conversation revolves around a student's math homework where they were tasked with creating a real-life scenario that can produce a sinusoidal function. The student's scenario involves a person bouncing on a trampoline and the height above the reference level being the dependent variable. The student's teacher argued that the graph could not be a sinusoidal function, but the student believes it is possible as long as the person stays in contact with the trampoline. However, the student's scenario is incomplete and they need to determine the minimum value of h when the person dips below the reference level. The student provides their own values for mass, force constant, and displacement, and is seeking help with equations to solve the problem.
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Jalhalla
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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
 
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  • #2
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.
 
  • #3
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.
 
Last edited:

1. What is the trampoline problem and why is it important?

The trampoline problem involves finding the minimum h value for a sinusoidal function that represents the height of a trampoline at a given time. This is important because it allows us to determine the maximum safe weight limit for the trampoline, ensuring it does not break or cause injury.

2. How do you approach solving the trampoline problem?

The trampoline problem can be solved by setting up a mathematical equation that represents the height of the trampoline as a function of time. This equation can then be graphed to visually determine the minimum h value. Alternatively, calculus techniques such as derivatives can be used to find the critical points of the function.

3. What factors affect the minimum h value in the trampoline problem?

The main factors that affect the minimum h value are the amplitude and frequency of the sinusoidal function, as well as the weight of the person jumping on the trampoline. A higher amplitude or frequency will result in a higher minimum h value, while a heavier weight will require a lower minimum h value for safety.

4. Are there any real-life applications for the trampoline problem?

Yes, the trampoline problem has many real-life applications, such as determining weight limits for amusement park rides, calculating safe load limits for bridges, and designing shock-absorbing materials for sports equipment.

5. Are there any limitations to the trampoline problem and its solutions?

One limitation of the trampoline problem is that it assumes a perfect sinusoidal function for the trampoline's height. In reality, the shape of the trampoline's surface may not be exactly sinusoidal, which can affect the accuracy of the minimum h value. Additionally, factors such as wind and uneven weight distribution can also impact the trampoline's performance and safety.

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