Astring of length l, mass per unit length \mu, and tension t is vibrating at its fundamental frequency. what effect will the following have on the fundamental frequency?
(a) the length of the string is doubled, with all other factors held constant ?
(b) the mass per unit length is doubled, with all other factors held constant.
(c) the tension is doubled, with all other factors held constant.

The fundamental frequency on a vibrating string is given by:

where

L is the length of the string

T is the tension

is the mass per unit length of the string

Keeping this equation in mind, we can now answer the various parts of the question:

(a) The fundamental frequency will halve

In this case, the length of the string is doubled:

L' = 2L

Substituting into the expression of the fundamental frequency, we find the new frequency:

So, the fundamental frequency will halve.

(b) the fundamental frequency will decrease by a factor

In this case, the mass per unit length is doubled:

Substituting into the expression of the fundamental frequency, we find the new frequency:

So, the fundamental frequency will decrease by a factor .

(c) the fundamental frequency will increase by a factor

In this case, the tension is doubled:

Substituting into the expression of the fundamental frequency, we find the new frequency:

So, the fundamental frequency will increase by a factor .

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