Young's Modulus and resonance frequency

Question

While trying to find an idea for a simple undergraduate level lab project (using PASCO sensors), I stumbled upon this equation:

$$Y = \frac{38.3*f_0^2*\rho*\ell^4 }{ d^2}$$

where: $Y$ = Young's Modulus

$f_0$ = resonance frequency of oscilations.

$\rho$ = density

$\ell$ = length

$d$ = width

I can't understand why Young's Modulus and the resonance frequency are related, and I can't seem to find anything helpful about it around the web. Could this be an empirical result..?

Thanks in advance

Answer 1

It's a real result. If a material has a high Young's modulus, it is "stiff". And with "stiffness" comes a strong force that tries to restore equilibrium for a certain displacement. In fact, the Young's modulus is directly proportional to the spring constant $k$ for a given geometry. So if you accept that a stiffer spring will give rise to a higher resonance frequency, you should be ready to accept the same for Young's modulus (all other things being equal).