To determine the frequency at which a 2.00-meter-long string will create standing waves with 3.00 loops, given that the wave speed is [tex]\(12.9 \, \text{m/s}\)[/tex], follow these steps:
### Step 1: Calculate the Wavelength
First, understand that for standing waves, the relationship between the string length [tex]\( L \)[/tex], the number of loops (or antinodes) [tex]\( n \)[/tex], and the wavelength [tex]\( \lambda \)[/tex] is given by:
[tex]\[
\lambda = \frac{2L}{n}
\][/tex]
where [tex]\( L \)[/tex] is the length of the string and [tex]\( n \)[/tex] is the number of loops.
Given:
- Length of the string, [tex]\( L = 2.00 \, \text{m} \)[/tex]
- Number of loops, [tex]\( n = 3.00 \)[/tex]
Plug in the values to find the wavelength:
[tex]\[
\lambda = \frac{2 \times 2.00 \, \text{m}}{3.00} = \frac{4.00 \, \text{m}}{3.00} = 1.3333 \, \text{m}
\][/tex]
So, the wavelength [tex]\( \lambda \)[/tex] is approximately [tex]\( 1.3333 \, \text{m} \)[/tex].
### Step 2: Calculate the Frequency
The frequency [tex]\( f \)[/tex] of the wave is related to the wave speed [tex]\( v \)[/tex] and the wavelength [tex]\( \lambda \)[/tex] by the equation:
[tex]\[
f = \frac{v}{\lambda}
\][/tex]
Given:
- Wave speed, [tex]\( v = 12.9 \, \text{m/s} \)[/tex]
- Wavelength, [tex]\( \lambda = 1.3333 \, \text{m} \)[/tex]
Using these values to find the frequency:
[tex]\[
f = \frac{12.9 \, \text{m/s}}{1.3333 \, \text{m}} \approx 9.675 \, \text{Hz}
\][/tex]
Therefore, the frequency at which the string will create standing waves with 3.00 loops is approximately [tex]\( 9.675 \, \text{Hz} \)[/tex].
