How is the frequency of a mass-spring system calculated?

The frequency of a mass-spring system is calculated using the formula f = 1/(2π) √(k/m), where k is the spring constant and m is the mass.

To understand this formula, we need to know what frequency means in the context of a mass-spring system. Frequency refers to the number of oscillations or cycles that the system completes in one second. It is measured in Hertz (Hz).

The frequency of a mass-spring system depends on two factors: the stiffness of the spring (k) and the mass (m) attached to it. A stiffer spring will have a higher frequency, while a heavier mass will have a lower frequency.

The formula for frequency takes into account both of these factors. The square root of k/m gives us the natural frequency of the system, which is the frequency at which it would oscillate if there were no external forces acting on it. However, in reality, there are always external forces (such as friction) that dampen the oscillations and reduce the frequency.

To calculate the actual frequency of the system, we need to divide the natural frequency by 2π. This factor accounts for the fact that the system completes one cycle when the displacement of the mass is equal to one wavelength, which is equal to 2π times the amplitude of the oscillation.

Let's look at an example. Suppose we have a mass-spring system with a spring constant of 10 N/m and a mass of 0.5 kg. Using the formula f = 1/(2π) √(k/m), we can calculate the frequency as follows:

f = 1/(2π) √(10/0.5)
f = 1/(2π) √20
f ≈ 1.26 Hz

Therefore, the frequency of the system is approximately 1.26 Hz.