Getting your Trinity Audio player ready...

Hold down the T key for 3 seconds to activate the audio accessibility mode, at which point you can click the K key to pause and resume audio. Useful for the Check Your Understanding and See Answers.

Period and Frequency of a Pendulum - help14

In most physical systems, changing one quantity will affect another quantity. And such it is with a vibrating pendulum. Changes in the length have an effect upon the period and the frequency of the pendulum motion. And the quantitative nature of the effect follows a very predictable pattern. Learn more about the pattern in the How to Think About This Situation section.

There are four similar versions of this question. Here is one of the versions:

Version 1:

Noah Formula is conducting an experimental study of the period of a pendulum. If Noah increases the mass of the string by a factor of two, then he can expect the period of the pendulum to …
a. increase by a factor of 2.
b. increase by a factor of 4.
c. increase by a factor of the square root of 2.
d. decrease by a factor of 2.
e. decrease by a factor of 4.
f. decrease by a factor of the square root of 2.
g. not be affected by the mass change.

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 one variable that affects the period of a pendulum is the length of the string that the pendulum bob is tied to. A longer string results in a longer period and a smaller frequency. That is, a bob on a long string will take a longer time to vibrate back and forth; the frequency at which it does its vibrations is not as frequent as a short string pendulum. The mass of the bob has little observable influence upon the period. The same is true of the angle from which the bob is released.

The Quantitative Relationship
The period (T) of a pendulum is directly proportional to the square root of the length (L) of the string from which the pendulum bob is supsended. The proportionality statement is ...

T ∝ √L

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

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

Using an Equation as a Guide to Thinking
Now you really have to think about the last bullet pooint above if you are to be successful in this third activity. It states that "the factor by which the period is changed is the square root of the factor by which the length is changed." That is to say, ...

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

What About Frequency?
The quantity frequency is the reciprocal of the period. And because it is, an increase in the period will cause the frequency to decrease. The two quantities are inversely proportional. So if this mathematical relationship is combined with the previous section, we can make the following claims:

an increase in the length causes the frequency to decrease,
a decrease in the length causes the frequency to increase,
the factor by which the frequency is increased is the square root of the factor by which the length is decreased, and
the factor by which the frequency is decreased is the square root of the factor by which the length is increased.

Try this link to The Physics Classroom Tutorial for more help with the concept of pendulum motion:

Pendulum Motion