Chapter 1.10, Problem 73E

(a)

To determine

The frequency of the string that is four times the tension.

Answer to Problem 73E

The frequency will be double the initial frequency

Explanation of Solution

Given information:

The fundamental frequency in hertz of a piano string is directly proportional to be square root of its tension and inversely proportional to its length and the square root of its mass density. A string gas the frequency of 100Hz.

Formula used:

  $f=\frac{k\sqrt{T}}{l\sqrt{\rho }}$

Calculation:

Find the frequency of a string with each property four times the tension, twice the length and four times the tension and twice the length.

The fundamental frequency $f$ of a piano string is directly proportional the square root of its tension T and inversely proportional to its length $l$ and the square root of its mass density So, the model will be $f=\frac{k\sqrt{T}}{l\sqrt{\rho }}$

When the tension is four times $T_1=4T$ , then the frequency of a string will be

  $\begin{array}{l}{f}_1=\frac{k\sqrt{T_1}}{l\sqrt{\rho }}\\ f_1=\frac{k\sqrt{4T}}{l\sqrt{\rho }}\\ f_1=2\left(\frac{k\sqrt{T}}{l\sqrt{\rho }}\right)\\ f_1=2f\end{array}$

Thus, the frequency will be double the initial frequency.

Conclusion:

The frequency will be double the initial frequency.

(b)

To determine

The frequency of the string is twice the length.

Answer to Problem 73E

The frequency will be half the initial frequency

Explanation of Solution

Given information:

The fundamental frequency in hertz of a piano string is directly proportional to be square root of its tension and inversely proportional to its length and the square root of its mass density. A string gas the frequency of 100Hz.

Formula used:

  $f=\frac{k\sqrt{T}}{l\sqrt{\rho }}$

Calculation:

When the length is twice $l_1=2l$ then the frequency of a string will be

  $\begin{array}{l}{f}_2=\frac{k\sqrt{T}}{l_1\sqrt{\rho }}\\ f_2=\frac{k\sqrt{T}}{2l\sqrt{\rho }}\\ f_2=\frac{1}{2}\left(\frac{k\sqrt{T}}{l\sqrt{\rho }}\right)\\ f_2=\frac{f}{2}\end{array}$

Thus, the frequency will be half the initial frequency.

Conclusion:

The frequency will be half the initial frequency

(c)

To determine

The frequency four times the tension and twice the length.

Answer to Problem 73E

The frequency will be same the initial frequency.

Explanation of Solution

Given information:

The fundamental frequency in hertz of a piano string is directly proportional to be square root of its tension and inversely proportional to its length and the square root of its mass density. A string gas the frequency of 100Hz.

Formula used:

  $f=\frac{k\sqrt{T}}{l\sqrt{\rho }}$

Calculation:

When the tension is four times $T_1=4T$ and twice the length $l_1=2l$ , then the frequency of a string will be

  $\begin{array}{l}{f}_3=\frac{k\sqrt{T_1}}{l_1\sqrt{\rho }}\\ f_3=\frac{k\sqrt{4T}}{2l\sqrt{\rho }}\\ f_3=\frac{2}{2}\left(\frac{k\sqrt{T}}{l\sqrt{\rho }}\right)\\ f_1=f\end{array}$

Thus, the frequency will be same the initial frequency.

Conclusion:

The frequency will be same the initial frequency.