Chapter 11, Problem 54P

To determine

To Find: The three other frequencies of string.

Answer to Problem 54P

The three other frequencies at which the particularstring resonates is $f_1=70\text{ Hz, }{f}_2=140\text{ Hz, }{f}_3=210\text{ Hz, }{f}_5=350\text{ Hz, }{f}_6=420\text{ Hz}$ .

Explanation of Solution

Given info:

String resonates in four loops at frequency, $f=280\text{ Hz}$

Formula used:

Write the expression for frequency for a nth string.

  $f_n=n\times f$ .

Calculation:

There are 4 loops, so frequency $f_4=4\times f_1\text{ = 280}$

  $\begin{array}{l}{f}_4=4\times f_1=280\\ \text{so, }4\times f_1=280\\ f_1=70\text{ Hz}\end{array}$

The three other frequencies at which the particular string resonates is,

  $\begin{array}{l}{f}_1=70\text{ Hz}\\ f_2=2\times 70=140\text{ Hz}\\ f_3=3\times 70=210\text{ Hz}\\ f_5=5\times 70=350\text{ Hz}\\ f_6=6\times 70=420\text{ Hz}\end{array}$

Conclusion:

Thus, the three other frequencies at which the particular string resonates is $f_1=70\text{ Hz, }{f}_2=140\text{ Hz, }{f}_3=210\text{ Hz, }{f}_5=350\text{ Hz, }{f}_6=420\text{ Hz}$ .