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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.8, Problem 28E

(a)

(i)

To determine

**To find:** The rate of change of the frequency with respect to the length (When

Expert Solution

The rate of change of frequency with respect to length is

**Given:**

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

**Calculation:**

Calculate the rate of change of frequency with respect to length when

Differentiate the equation (1) with respect to

Thus, the rate of change of frequency with respect to length is

(i)

To determine

**To find:** The rate of change of the frequency with respect to the length (When

Expert Solution

The rate of change of frequency with respect to length is

**Given:**

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

**Calculation:**

Calculate the rate of change of frequency with respect to length when

Differentiate the equation (1) with respect to

Thus, the rate of change of frequency with respect to length is

(ii)

To determine

**To find:** The rate of change of the frequency with respect to the tension (When

Expert Solution

The rate of change of frequency with respect to tension is

Calculate the rate of change of frequency with respect to tension when

Differentiate the equation (1) with respect to

Thus, the rate of change of frequency with respect to tension is

(iii)

To determine

**To find:** The rate of change of the frequency with respect to the linear density (When

Expert Solution

The rate of change of frequency with respect to linear density is

Calculate the rate of change of frequency with respect to linear density when

Differentiate the equation (1) with respect to

Thus, the rate of change of frequency with respect to linear density is

(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”.

Expert Solution

The pitch of the note is higher.

Determine the behavior of the pitch note when the effective length of a string is decreased.

Sketch the curve.

Calculate the value of

Here

Substitute 1 for

Similarly, calculate the remaining values of

Calculate and tabulate the remaining values of

1 | 2 | 0.5 |

1 | 3 | 0.3 |

1 | 4 | 0.25 |

1 | 5 | 0.2 |

1 | 6 | 0.17 |

1 | 7 | 0.14 |

1 | 8 | 0.12 |

1 | 9 | 0.11 |

1 | 10 | 0.1 |

1 | 11 | 0.09 |

**Graph:**

Plot the graph using table (1) as shown in figure (1).

Comments:

Refer the figure (1).

Here the frequency of vibration

(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”.

Expert Solution

The pitch of the note is higher.

Determine the behavior of the pitch note when the effective length of a string is decreased.

Sketch the curve.

Calculate the value of

Here

Substitute 1 for

Similarly, calculate the remaining values of

Calculate and tabulate the remaining values of

1 | 2 | 0.5 |

1 | 3 | 0.3 |

1 | 4 | 0.25 |

1 | 5 | 0.2 |

1 | 6 | 0.17 |

1 | 7 | 0.14 |

1 | 8 | 0.12 |

1 | 9 | 0.11 |

1 | 10 | 0.1 |

1 | 11 | 0.09 |

**Graph:**

Plot the graph using table (1) as shown in figure (1).

Comments:

Refer the figure (1).

Here the frequency of vibration

(ii)

To determine

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

Expert Solution

The pitch of the note is higher.

Determine the behavior of the pitch note when the tension is increased by turning a tuning peg.

Sketch the curve.

Calculate the value of

Here

Substitute 1 for

Similarly, calculate the remaining values of

Calculate and tabulate the remaining values of

1 | 0.0 | 0.0 |

1 | 0.2 | 0.4 |

1 | 0.4 | 0.6 |

1 | 0.6 | 0.8 |

1 | 0.8 | 0.9 |

1 | 1.0 | 1.0 |

1 | 1.2 | 1.1 |

1 | 1.4 | 1.2 |

1 | 1.6 | 1.3 |

1 | 1.8 | 1.3 |

1 | 2.0 | 1.4 |

1 | 2.2 | 1.5 |

1 | 2.4 | 1.5 |

1 | 2.6 | 1.6 |

1 | 2.8 | 1.7 |

1 | 3.0 | 1.7 |

1 | 3.2 | 1.8 |

1 | 3.4 | 1.8 |

1 | 3.6 | 1.9 |

1 | 3.8 | 1.9 |

1 | 4.0 | 2.0 |

1 | 4.2 | 2.0 |

1 | 4.4 | 2.1 |

1 | 4.6 | 2.1 |

1 | 4.8 | 2.2 |

1 | 5.0 | 2.2 |

1 | 5.2 | 2.3 |

1 | 5.4 | 2.3 |

1 | 5.6 | 2.4 |

1 | 5.8 | 2.4 |

1 | 6.0 | 2.4 |

1 | 6.2 | 2.5 |

1 | 6.4 | 2.5 |

1 | 6.6 | 2.6 |

1 | 6.8 | 2.6 |

1 | 7.0 | 2.6 |

1 | 7.2 | 2.7 |

1 | 7.4 | 2.7 |

1 | 7.6 | 2.8 |

1 | 7.8 | 2.8 |

1 | 8.0 | 2.8 |

1 | 8.2 | 2.9 |

1 | 8.4 | 2.9 |

1 | 8.6 | 2.9 |

1 | 8.8 | 3.0 |

1 | 9.0 | 3.0 |

1 | 9.2 | 3.0 |

1 | 9.4 | 3.1 |

1 | 9.6 | 3.1 |

1 | 9.8 | 3.1 |

1 | 10.0 | 3.2 |

Graph:

Plot the graph using table (2) shown in figure (2).

Comments:

Refer the figure (2).

Here the frequency of vibration

(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”.

Expert Solution

The pitch of the note is lower.

Determine the behavior of the pitch note when the linear density is increased by switching to another string.

Sketch the curve.

Calculate the value of

Here

Substitute 1 for

Similarly, calculate the remaining values of

Calculate and tabulate the remaining values of

1 | 0.2 | 2.2 |

1 | 0.4 | 1.6 |

1 | 0.6 | 1.3 |

1 | 0.8 | 1.1 |

1 | 1.0 | 1.0 |

1 | 1.2 | 0.9 |

1 | 1.4 | 0.8 |

1 | 1.6 | 0.8 |

1 | 1.8 | 0.7 |

1 | 2.0 | 0.7 |

1 | 2.2 | 0.7 |

1 | 2.4 | 0.6 |

1 | 2.6 | 0.6 |

1 | 2.8 | 0.6 |

1 | 3.0 | 0.6 |

1 | 3.2 | 0.6 |

1 | 3.4 | 0.5 |

1 | 3.6 | 0.5 |

1 | 3.8 | 0.5 |

1 | 4.0 | 0.5 |

1 | 4.2 | 0.5 |

1 | 4.4 | 0.5 |

1 | 4.6 | 0.5 |

1 | 4.8 | 0.5 |

1 | 5.0 | 0.4 |

1 | 5.2 | 0.4 |

1 | 5.4 | 0.4 |

1 | 5.6 | 0.4 |

1 | 5.8 | 0.4 |

1 | 6.0 | 0.4 |

1 | 6.2 | 0.4 |

1 | 6.4 | 0.4 |

1 | 6.6 | 0.4 |

1 | 6.8 | 0.4 |

1 | 7.0 | 0.4 |

1 | 7.2 | 0.4 |

1 | 7.4 | 0.4 |

1 | 7.6 | 0.4 |

1 | 7.8 | 0.4 |

1 | 8.0 | 0.4 |

1 | 8.2 | 0.3 |

1 | 8.4 | 0.3 |

1 | 8.6 | 0.3 |

1 | 8.8 | 0.3 |

1 | 9.0 | 0.3 |

1 | 9.2 | 0.3 |

1 | 9.4 | 0.3 |

1 | 9.6 | 0.3 |

1 | 9.8 | 0.3 |

1 | 10.0 | 0.3 |

Graph:

Plot the graph using table (3) as shown in figure (3).

Comments:

Refer the figure (3).

Here the frequency of vibration