Respuesta :
Answer:
the length of the string L is approximately 4.257 meters.
Explanation:
To solve this problem, we can use the formula for the wavelength of a harmonic in a string fixed at both ends:
[tex][tex]\[ \lambda_n = \frac{2L}{n} \][/tex][/tex]
where:
[tex]- \( \lambda_n \) is the wavelength of the nth harmonic,[/tex]
[tex]- \( L \) is the length of the string,[/tex]
[tex]- \( n \) is the harmonic number.[/tex]
Given that the wavelength of the nth harmonic is 0.4714 m and the wavelength of the (n−1)th harmonic is 0.5300 m, we can set up the following equations:
[tex]\[ \frac{2L}{n} = 0.4714 \][/tex]
[tex]\[ \frac{2L}{n-1} = 0.5300 \][/tex]
Now, we can solve these equations simultaneously to find the value of L.
First, rearrange the equations:
[tex]\[ L = \frac{n}{2} \cdot 0.4714 \][/tex]
[tex]\[ L = \frac{n-1}{2} \cdot 0.5300 \][/tex]
Now, equate the two expressions for L:
[tex]\[ \frac{n}{2} \cdot 0.4714 = \frac{n-1}{2} \cdot 0.5300 \][/tex]
Now, solve for L:
[tex]\[ n \cdot 0.4714 = (n-1) \cdot 0.5300 \][/tex]
[tex]\[ 0.4714n = 0.5300n - 0.5300 \][/tex]
[tex]\[ 0.5300 - 0.4714n = 0 \][/tex]
[tex]\[ 0.0586n = 0.5300 \][/tex]
[tex]\[ n = \frac{0.5300}{0.0586} \][/tex]
[tex]\[ n \approx 9.03 \][/tex]
Since n must be a positive integer, we can take the integer part of [tex]\( 9.03 \), which is \( 9 \).[/tex]
Now, substitute n = 9 into either of the original equations to find L:
[tex]\[ L = \frac{9}{2} \cdot 0.4714 \][/tex]
[tex]\[ L = 4.257 \][/tex]
So, the length of the string L is approximately 4.257 meters.
Final answer:
The length of a string, given the wavelengths of its nth and (n-1)th harmonics, is determined using the formula relating wavelength and string length. By applying the given wavelengths, we find the string length to be approximately 2.12 meters.
Explanation:
The question asks us to find the length of a string when given the wavelengths of its nth and (n-1)th harmonics. The formula that relates the wavelength of standing waves on a string to the length of the string is λ = 2L/n, where λ is the wavelength, L is the length of the string, and n is the harmonic number. Given the wavelengths for the nth harmonic (0.4714 m) and the (n-1)th harmonic (0.5300 m), we can set up two equations based on this formula for n and n-1 harmonics and solve for L.
To simplify, let's assign λ1 = 0.4714 m for the nth harmonic and λ2 = 0.5300 m for the (n-1)th harmonic. The equations become: λ1 = 2L/n and λ2 = 2L/(n-1). Dividing these two equations, we get λ1 /λ2 = n/(n-1). Substituting the given values, 0.4714 / 0.5300 = n/(n-1). Solving this for n gives n ≈ 8.86. Since n must be an integer, n is approximately equal to 9. Re-substituting into either equation for λ and solving for L gives us the length of the string.
Using λ1 and n = 9, L = (0.4714 m * 9) / 2 ≈ 2.12 m. Therefore, the length of the string is approximately 2.12 meters.
