Sure, let's walk through the problem step-by-step.
Given:
- The frequency of the 4th harmonic of a string is 31.5 Hz.
What's asked:
- To find the fundamental frequency of the string.
### Step-by-Step Solution:
1. Understanding Harmonics:
- The harmonic frequencies of a string are integer multiples of the fundamental frequency.
- This means the [tex]\( n \)[/tex]-th harmonic frequency is given by:
[tex]\[
f_n = n \cdot f_1
\][/tex]
where [tex]\( f_1 \)[/tex] is the fundamental frequency, [tex]\( f_n \)[/tex] is the frequency of the [tex]\( n \)[/tex]-th harmonic, and [tex]\( n \)[/tex] is the harmonic number.
2. Given Information:
- We know the 4th harmonic frequency ([tex]\( f_4 \)[/tex]) is 31.5 Hz.
- The harmonic number for this frequency ([tex]\( n \)[/tex]) is 4.
3. Express the 4th harmonic frequency in terms of the fundamental frequency:
[tex]\[
f_4 = 4 \cdot f_1
\][/tex]
4. Solve for the fundamental frequency ([tex]\( f_1 \)[/tex]):
- We can rearrange the equation to isolate [tex]\( f_1 \)[/tex]:
[tex]\[
f_1 = \frac{f_4}{4}
\][/tex]
- Plugging in the given value [tex]\( f_4 = 31.5 \)[/tex] Hz:
[tex]\[
f_1 = \frac{31.5}{4}
\][/tex]
5. Divide to find the fundamental frequency:
[tex]\[
f_1 = 7.875 \text{ Hz}
\][/tex]
So, the fundamental frequency of the string is:
[tex]\[
f_1 = 7.875 \text{ Hz}
\][/tex]
