When discussing springs a little terminology needs to be explained. The spring's equilibrium position is testing position of the mass when it is attached to the spring.

The amplitude is defined as the distance between the equilibrium position and the end of an oscillation. In the animation below the distance between A & B and B & C are the spring's two, equal, amplitudes.

When exhibiting simple harmonic motion an oscillating spring moves back-and-forth without any driving or damping forces acting on it.

In the real word all springs have damping forces acting on them. This is due to the fact that as the metal changes shape the molecules rub against each other. But in the short term they will behave as a perfect oscillating system.

The spring and mass system oscillate with a resonant or natural frequency. The oscillating motion of a spring is related to circular motion.

Notice how the maximum amplitude is the "x" in Hooke's law and "x" equals the disk's radius. Below is the derivation describing the spring's period of motion.

Where

• T is the period of motion. This the time to return to one of the maximum amplitudes or the other -measured in seconds. This animated spring above takes about 5 seconds to start on the left and return to the left position. It has a period of motion of about 5 s.
• "m" is the mass attached to the end of the spring -measured in kg.
• "k" is the same spring constant that is found in Hooke's law -measured in N/m.

The resonant frequency for a spring is found from

Where "T" is from the equation above.

Example 3

Question Hint Solution A spring is laying horizontally on a frictionless table. It is oscillating back-and-forth. It has an attached mass of 200 g and a spring constant of 50.0 N/m. What is the resonant frequency? List your givens. Strategy: Convert the mass to kg. Calculate the period. Use the period to calculate the resonant frequency.