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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The frequency of vibrations of a vibrating violin string is given by

F = 1 2 L T ρ

where L is the length of the string, T is its tension, and ρ is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA: Brooks/Cole, 2002).]

(a) Find the rate of change of the frequency with respect to

(i) the length (when T and ρ are constant),

(ii) the tension (when L and ρ are constant), and

(iii) the linear density (when L. and Tare constant).

(b) The pitch of a note (how high or low the note sounds) is determined by the frequency f. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note

(i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates,

(ii) when the tension is increased by turning a tuning peg.

(iii) when the linear density is increased by switching to another string.

(a)

(i)

To determine

To find: The rate of change of the frequency with respect to the length (When T and ρ are constant)

Explanation

Given:

The frequency of vibrations of a vibrating violin string is as given below.

f=12LTρ (1)

Calculation:

Calculate the rate of change of frequency with respect to length when T and ρ are constant.

Differentiate the equation (1) with respect to L.

dfdL=ddL(12LTρ)=12Tρ(d(L)1dL)

(i)

To determine

To find: The rate of change of the frequency with respect to the length (When T and ρ are constant)

(ii)

To determine

To find: The rate of change of the frequency with respect to the tension (When L and ρ are constant)

(iii)

To determine

To find: The rate of change of the frequency with respect to the linear density (When L and T are constant)

(b)

(i)

To determine

To find: To determine the behavior of pitch of the note “when the effective length of a string is decreased by placing a finger on the string is decreased by placing a finger on the string so a shorter portion of the string vibrates”.

(i)

To determine

To find: To determine the behavior of pitch of the note “when the effective length of a string is decreased by placing a finger on the string is decreased by placing a finger on the string so a shorter portion of the string vibrates”.

(ii)

To determine

To find: To determine the behavior of pitch of the note “When the tension is increased by turning a tuning peg.”.

(iii)

To determine

To find: To determine the behavior of pitch of the note “When the linear density is increased by switching to another string”.

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