In most physical systems, changing one quantity will affect another quantity. And such it is with vibrating masses attached to springs. Changes in the mass and the spring constant have an effect upon the period and the frequency of the vibrational motion. And the quantitative nature of these effects follow very predictable patterns. Learn more about the patterns in the How to Think About This Situation section.
​

Factors Affecting Period and Frequency
Perhaps you have done the experiment. It certainly is more exciting to discover the relationship on your own than to be told what it is. And if you have done the experiment, then you know that the two variables that affect the period of a vibrating mass on a spring are the mass of the object and the spring constant of the spring. A more massive object hung on the spring will vibrate with a longer period and a smaller frequency. Mass and period are directly related while mass and frequency are inversely related.  And a stronger (stiffer) spring with a larger spring constant results in a shorter period and a higher frequency. That is, an object on a stiff spring will take less time to vibrate back and forth; the frequency at which it does its vibrations will be higher than it would be for a weaker (less rigid) spring. Spring constant and period are inversely related while spring constant and frequency are directly related.

The formula for determining the period includes m/k as a ratio in the equation. In fact, the period (T) of a vibrating mass on a spring is directly proportional to the square root of the m/k ratio of the mass/spring system. The proportionality statement is ...

T ∝ √(m/k)

The Quantitative Relationship
The period (T) of a vibrating mass on a spring is directly proportional to the square root of the mass (m) and inversely proportional to the square root of the spring constant (k) of the spring. The proportionality statement is ...

T ∝ √(m/k)

To say that period is proportional to the square root of mass means that ...

• an increase in the mass causes the period to increase,
• a decrease in the mass causes the period to decrease, and
• the factor by which the period is changed is the square root of the factor by which the mass is changed.

And to say that period is inversely proportional to the square root of spring constant means that ...

• an increase in the spring constant causes the period to decrease,
• a decrease in the spring constant causes the period to increase, and
• the factor by which the period is changed is the reciprocal of the square root of the factor by which the spring constant is changed.

Using an Equation as a Guide to Thinking
Now you really have to think about the last pair of bullet points above if you are to be successful in this third activity. Let's start with period and mass. It states that "the factor by which the period is changed is the square root of the factor by which the mass is changed." That is to say, ...

• if the mass is increased by a factor of 2, the period will be increased by a factor of √2.
• if the mass is decreased by a factor of 2, the period will be decreased by a factor of √2.
• if the mass is increased by a factor of 3, the period will be increased by a factor of √3.
• if the mass is decreased by a factor of 3, the period will be decreased by a factor of √3.
• if the mass is increased by a factor of 4, the period will be increased by a factor of √4.
• if the mass is decreased by a factor of 4, the period will be decreased by a factor of √4.

• if the spring constant is increased by a factor of 2, the period will be decreased by a factor of √2.
• if the spring constant is decreased by a factor of 2, the period will be increased by a factor of √2.
• if the spring constant is increased by a factor of 3, the period will be decreased by a factor of √3.
• if the spring constant is decreased by a factor of 3, the period will be increased by a factor of √3.
• if the spring constant is increased by a factor of 4, the period will be decreased by a factor of √4.
• if the spring constant is decreased by a factor of 4, the period will be increased by a factor of √4.