How to find Young's elastic modulus of a sphere using a camera? So I have a deformable ball and a high-speed camera. How do I measure Young's elastic modulus?
I was thinking of looking at the coefficient of restitution of the ball, but it doesn't seem to be directly related to the elastic modulus, to my surprise (I might be wrong, though). Alternatively, I could use Hertzian contact laws, but this gives me the corrected elastic modulus, which also contains Poisson's ratio, which I do not have. I have only one ball size.
Do you have any suggestions?
 A: I think that you could use the Hertzian contact problem for this.
Let's say that you have a plane surface with a Young modulus and a mass that are much higher than the ones from your sphere. You can use the equation
$$h_0 = \left(\frac{m}{k}\right)^{2/5} v^{4/5}\, ,$$
with $m$ the mass of your sphere, and $v$ its speed. Here, $h_0$ represents the deformation when the sphere starts to bounce back. If your sphere is much more compliant than your surface you can assume that all the deformation is happening in that object. The speed is the value you achieve just before impact, so you can vary it with different initial heights. Actually, the equation above is obtained from a balance of energy, so you could rederive it for different heights instead of speeds (see reference).
The other parameters are:
$$k = \frac{4}{5D}\sqrt{R}\, ,$$
with $R$ the radius of the sphere and
$$D = \frac{3}{4}\frac{(1 - \nu^2)}{E}\, .$$
You could vary $v$ and then perform a nonlinear regression to find $\nu$ and $E$.
Reference

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*Landau, L. D., & Lifshitz, E. M. (1986). Theory of elasticity (Vol. 7, No. 3). New York: Pergamon Press, Oxford.

