To solve for the new frequency that would create 7.00 loops, we'll use the relationship between the frequency and the number of loops in a standing wave. The frequency of a vibrating string is directly proportional to the number of loops.
We start with the given data:
- The original frequency is 60.0 Hz.
- The number of loops originally is 5.00.
- The new number of loops is 7.00.
Since the frequency and the number of loops are proportional, we can set up a proportion to find the new frequency. This relationship can be stated mathematically as:
[tex]\[
\frac{f_{\text{original}}}{L_{\text{original}}} = \frac{f_{\text{new}}}{L_{\text{new}}}
\][/tex]
Where:
- [tex]\( f_{\text{original}} \)[/tex] is the original frequency (60.0 Hz).
- [tex]\( L_{\text{original}} \)[/tex] is the original number of loops (5.00).
- [tex]\( f_{\text{new}} \)[/tex] is the new frequency we need to find.
- [tex]\( L_{\text{new}} \)[/tex] is the new number of loops (7.00).
We need to rearrange this proportion to solve for [tex]\( f_{\text{new}} \)[/tex]:
[tex]\[
f_{\text{new}} = f_{\text{original}} \times \frac{L_{\text{new}}}{L_{\text{original}}}
\][/tex]
Substituting the known values:
[tex]\[
f_{\text{new}} = 60.0 \, \text{Hz} \times \frac{7.00}{5.00}
\][/tex]
Next, we perform the division and multiplication:
[tex]\[
\frac{7.00}{5.00} = 1.4
\][/tex]
[tex]\[
f_{\text{new}} = 60.0 \, \text{Hz} \times 1.4 = 84.0 \, \text{Hz}
\][/tex]
Thus, the new frequency that would create 7.00 loops is:
[tex]\[
f_{\text{new}} = 84.0 \, \text{Hz}
\][/tex]
