The frequency of a mass-spring system is determined using the formula

where $k$ represents the spring constant and $m$ denotes the mass attached to the spring.

To grasp this formula, it’s important to understand what frequency signifies in the context of a mass-spring system. Frequency refers to the number of complete oscillations or cycles that the system undergoes in one second, and it is measured in Hertz (Hz).

The frequency of a mass-spring system is influenced by two main factors: the stiffness of the spring, denoted by $k$, and the mass $m$ that is connected to it. A stiffer spring results in a higher frequency, while an increased mass leads to a lower frequency.

The frequency formula incorporates both factors. The term $\sqrt{\frac{k}{m}}$ yields the natural frequency of the system — the frequency at which the system would oscillate in the absence of any external forces. However, in practical scenarios, external forces, such as friction, invariably dampen the oscillations and consequently reduce the frequency.

To compute the actual frequency of the system, we divide the natural frequency by $2\pi$. This division accounts for the fact that one complete cycle occurs when the displacement of the mass equals one wavelength, which is equal to $2\pi$ times the amplitude of oscillation.

Consider an example: suppose we have a mass-spring system with a spring constant of $10 \, \text{N/m}$ and a mass of $0.5 \, \text{kg}$. We can calculate the frequency using the formula

Calculating this step by step:

1. First, compute $\frac{10}{0.5} = 20$.
2. Next, take the square root: $\sqrt{20} \approx 4.47$.
3. Finally, divide by $2\pi$:

Therefore, the frequency of the system is approximately $0.71 \, \text{Hz}$.