When a mass \(M\) oscillates on a spring having its own mass \(m\) and stiffness \(k\), the natural frequency of the system is affected by the combined effective mass of both the attached mass and the spring itself. Understanding this system requires considering the spring's contribution to the total kinetic energy.

Natural Frequency Basics

The natural frequency is the inherent frequency at which a system vibrates when disturbed and then allowed to oscillate freely. For an ideal, massless spring with an attached mass \(M\), the natural angular frequency \(\omega_n\) is given by the formula:

\[\omega_n = \sqrt{\frac{k}{M}}\]

Here:

• \(k\) represents the stiffness (or spring constant) of the spring, measured in Newtons per meter (\(N/m\)).
• \(M\) represents the mass attached to the spring, measured in kilograms (\(kg\)).

This fundamental formula applies when the spring's mass is considered negligible.

Spring Mass Contribution to Oscillation

In many real-world scenarios, the mass \(m\) of the spring is not negligible and plays a role in the system's dynamics. As the mass \(M\) oscillates, different parts of the spring move with varying velocities. The end of the spring attached to the mass \(M\) moves with the same velocity as \(M\), while the fixed end of the spring has zero velocity.

To account for the spring's distributed mass in the oscillation, we determine an "effective mass" for the spring. This effective mass represents the portion of the spring's mass that, if concentrated at the end of a massless spring, would result in the same kinetic energy as the actual oscillating spring. For a uniform spring, it has been derived that the effective mass \(m_{eff}\) that should be added to the oscillating mass is one-third of the spring's total mass:

\[m_{eff} = \frac{m}{3}\]

This \(m_{eff}\) is derived by integrating the kinetic energy of infinitesimal segments of the spring along its length.

System Total Effective Mass

To calculate the natural frequency of the entire system, we need to consider the total effective mass that is participating in the oscillation. This total effective mass, often denoted as \(M_{total}\), is the sum of the attached mass \(M\) and the effective mass of the spring \(m_{eff}\):

\[M_{total} = M + m_{eff} = M + \frac{m}{3}\]

This \(M_{total}\) now replaces the single mass \(M\) in the standard natural frequency formula, providing a more accurate representation of the system.

Natural Frequency Calculation

Using the total effective mass, the natural frequency \(\omega_n\) of the system where mass \(M\) oscillates on a spring with mass \(m\) and stiffness \(k\) can be expressed as:

\[\omega_n = \sqrt{\frac{k}{M_{total}}}\]

Substituting the expression for \(M_{total}\) into the formula:

\[\omega_n = \sqrt{\frac{k}{M+\frac{m}{3}}}\]

This formula precisely describes the natural frequency of the given system, taking into account the mass of the spring.

Options Comparison

Let's evaluate the provided options based on our derived formula:

• Option 1: \(\frac k{M}\) - This expression is dimensionally incorrect for frequency (it would represent \(\omega_n^2\)) and does not account for the spring's mass.
• Option 2: \(\frac km\) - This expression is also dimensionally incorrect for frequency and considers only the spring's mass, not the oscillating mass \(M\).
• Option 3: \(\sqrt{\frac{k}{M+\frac{{{m}}}{3}}} \) - This option perfectly matches our derived formula, including the effective mass of the spring and the correct square root for frequency.
• Option 4: \(\sqrt{\frac{k}{M+{{{m}}}} \) - This option incorrectly assumes that the entire mass of the spring contributes directly to the oscillating mass, rather than just one-third of it.

Based on the principles of oscillation and the effective mass of a spring, the correct natural frequency for the system is given by \(\sqrt{\frac{k}{M+\frac{{{m}}}{3}}} \).