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How do you determine energy transferred to a trampoline when a ball is dropped on it?

Physics Asked by user266637 on August 20, 2020

Suppose a ball of mass $m$ is dropped over a trampoline(consider the young’s modulus of the material to be $Y$) from some height $h$. The ball impacts the trampoline, gets slowed down due to the restoring force generated in the trampoline as it curves down, and finally comes to a stop to again begin accelerating in the opposite direction and leave the system after rebounding. What happens in this process is that the ball’s energy(K.E) does not remain same as before. We can expect a bit of vibrational energy being transferred to the trampoline in this process. Is there any way to calculate(theoretically) how much vibrational energy is being transferred to the trampoline. Consider other energy losses as negligible.

One Answer

Being physicists, we're going to start with a 1 dimensional trampoline, which is a mass $M$, on a spring with constant $k$, all damped ($c$), so that:

$$ Ma = F $$ $$ Mddot x = (- cdot x - k x) + F(t) $$

where $x=0$ is the equilibrium position, and $F(t)$ is an external driving force.

That can be rewritten as:

$$ ddot x(t) + 2 xi omega_0 dot x(t) + omega_0^2 x(t) = F(t)/M $$

where $omega_0 = sqrt{k/M}$ is the undamped angular frequency and $xi = frac c 2 frac 1 {sqrt{Mk}}$ the damping ratio.

Note that this is one of the most used equations in physics and engineering, so I'm not going to solve it here.

The question is now, how do you model the bounce?

I'd go with "a $pi$'s worth of $sin(omega_m t)$", or:

$$ F(t) = Asin(omega_m t) 0 le omega_m t le pildots $$

$0$ otherwise.

With

$$omega_m = sqrt{frac k {m+M}}$$

and

$$ frac 1 2 kA^2 = mgh $$

From here, you will get a frequency response function of the "trampoline" and then drive that with the frequency spectrum represented by $tilde F(omega)$, the Fourier transform of $F(t)$.

Answered by JEB on August 20, 2020

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