This question asks for the natural frequency of a simply supported beam with specific properties. Let's break down how to determine this.

Understanding Beam Natural Frequency

The natural frequency ($f$) of a vibrating system is the frequency at which it will oscillate if disturbed from its equilibrium position. For a simple mass-spring system, the natural frequency is given by $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, where $k$ is the stiffness of the spring and $m$ is the mass. In this case, the beam acts as the spring, and the mass $M$ at the center is the oscillating mass.

Calculating Beam Stiffness

To find the natural frequency, we first need to determine the effective stiffness ($k$) of the beam. The stiffness relates the applied force to the resulting static deflection ($\delta$). The formula is $k = \frac{P}{\delta}$, where $P$ is the applied force.

For a simply supported beam of length $l$, flexural rigidity $EI$, subjected to a concentrated load $P$ at its center, the maximum static deflection ($\delta$) occurs at the center and is given by the standard beam deflection formula:

\(\delta = \frac{P{{l}^{3}}}{48EI}\)

Now, we can find the stiffness ($k$) by rearranging this formula:

\({k} = \frac{P}{\delta} = \frac{P}{\frac{P{{l}^{3}}}{48EI}} = \frac{48EI}{{{l}^{3}}}\)

Determining the Natural Frequency

With the stiffness ($k$) and the mass ($M$) known, we can now calculate the natural frequency. The formula for natural frequency ($f$) is:

\({f} = \frac{1}{2\pi}\sqrt{\frac{k}{M}}\)

Substituting the expression for $k$ we found:

\({f} = \frac{1}{2\pi}\sqrt{\frac{\frac{48EI}{{{l}^{3}}}}{M}}\)

Simplifying this expression gives us the final formula for the natural frequency:

\({f} = \frac{1}{2\pi}\sqrt{\frac{48EI}{M{{l}^{3}}}}\)

Final Answer Check

Comparing this result with the given options, we find that it matches the expression provided in the first option.