A string that is fixed at both ends has a length of 2.36 m. When the string vibrates at a frequency of 77.0 Hz, a standing wave with five loops is formed. What is the wavelength of the waves that travel on the string?

Question

A string that is fixed at both ends has a length of 2.36 m. When the string vibrates at a frequency of 77.0 Hz, a standing wave with five loops is formed. What is the wavelength of the waves that travel on the string?

BlackTom AI · Accepted Answer

To analyze this problem, I start by identifying what the phrase “five loops” means for a string fixed at both ends. Each loop corresponds to half of a wavelength along the string, so the length L of the string equals n times (lambda/2), where n is the number of loops.

Option 1: 4.7 m
Using L = n(lambda/2) with n = 5, lambda would be lambda = 2L/n = 2*2.36/5 = 0.944 m, not 4.7 m. Thus this value is inconsistent with five loops on a 2.36 m string.

Option 2: 0.94 m
If lambda = 0.94 m, then L = n(lambda/2) = 5*(0.94/2) = 5*0.47 = 2.35 m, which is essentially the given length 2.36 m within rounding. This matches the condition of having five half-wavelengths along the string, so this is consistent.

Option 3: 2.2 m
A wavelength of 2.2 m would give L = 5*(2.2/2) = 5*1.1 = 5.5 m, which is far longer than the actual string length. This cannot be correct for a five-loop standing wave on a 2.36 m string.

Option 4: 1.5 m
With lambda = 1.5 m, L would be 5*(1.5/2) = 5*0.75 = 3.75 m, again not equal to 2.36 m. This shows the value is incompatible with the given length and loop count.

Option 5: 11 m
A wavelength of 11 m would yield L = 5*(11/2) = 5*5.5 = 27.5 m, which is nowhere near the actual string length. This clearly contradicts the setup of the problem.

In summary, the key concept is that, for a string fixed at both ends, the number of loops equals the number of half-wavelengths along the string, so L = n(lambda/2). Only lambda = 0.94 m satisfies the relationship with L = 2.36 m and n = 5, making it the consistent choice among the options.

2025 Fall PHYS 2205 TR 11:00 - 12:15 _ 89122 2205 Final Exam

A string that is fixed at both ends has a length of 2.36 m. When the string vibrates at a frequency of 77.0 Hz, a standing wave with five loops is formed. What is the wavelength of the waves that travel on the string?

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