Question

A cellist tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 m long and has a mass of 14.5 g

A.) With what tension must she stretch that portion of the string?
B.) What percentage increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from C to D?

Answer 1

Concepts and reason

The concepts used to solve this problem are fundamental frequency and tension on a string.

Initially, use the concept of fundamental frequency of the string to calculate the tension on the string.

Initially, use the relationship between frequency and tension of the string to find the relationship between the actual and current tension in the string.

Finally, use the relationship between percentage change in tension, actual tension, and changed tension to find the required percentage that increases in tension.

Fundamentals

The expression for the frequency on a string is as follows:

Here, frequency is , length of the string is , tension on the string is , and mass is .

The expression for the percentage change in tension is as follows:

Here, percentage change in tension is , and the changed tension is .

(A)

The expression for the fundamental actual frequency in the string is as follows:

Here, the fundamental actual frequency is , and the actual tension is .

Rearrange the expression for .

Substitute for , for , and for .

(B)

The expression for the fundamental actual frequency in the string is as follows:

…… (1)

Here, the fundamental actual frequency is and the actual tension is .

The expression for the fundamental current frequency in the string is as follows:

…… (2)

Here, the fundamental current frequency is and current tension is .

From Equations (1) and (2):

The ratio between current and actual tension is as follows:

Substitute for and for .

Substitute for .

The expression for the percentage increase in the tension is as follows:

Substitute for and for .

Ans: Part A

The tension in the string is .

Part B

The required percentage increase in the tension is .