A vibrating stretched string has length 31 cm, mass 19 grams and is under a tension of 38 newtons. What is the frequency of its 3rd harmonic?
If the source of the wave is an open organ pipe of the same length as the wire, what would be the frequency of the 4 th harmonic?
How about showing that you have made some effort of your own? Find the appropriate equation in your course materials instead of expecting someone else to do it for you. Someone will gladly critique your work.
The formula will require that you know the wave speed in the wire. That is
V = sqrt[(Tension)/(linear density)]
= sqrt[38N/(.019kg/.31m)] = 24.9 m/s
The harmonic number is determined by how many half-wavelengths there are in the wire's standing wave pattern. For the frequency, you will use
F*(wavelength) = V
To find the frequency of the 3rd harmonic, we can use the equation:
f = (nv)/(2L)
Where:
f = frequency of the harmonic
n = harmonic number
v = velocity of the wave
L = length of the vibrating string
First, let's calculate the velocity of the wave using the given information. The velocity of a wave can be calculated using the equation:
v = √(T/μ)
Where:
v = velocity of the wave
T = tension in the string
μ = mass per unit length
Given that the tension (T) is 38 newtons and the mass (m) is 19 grams, we need to convert the mass to mass per unit length (μ) by dividing it by the length of the string. Let's convert the mass to kilograms first:
19 grams = 0.019 kilograms
Now, let's calculate the mass per unit length:
μ = m/L = 0.019 kg / 0.31 m = 0.0613 kg/m
Next, we can calculate the velocity:
v = √(38 N / 0.0613 kg/m) = 49.19 m/s (approximately)
Now that we have the velocity, we can calculate the frequency of the 3rd harmonic:
f = (3 * v) / (2 * L) = (3 * 49.19 m/s) / (2 * 0.31 m) = 75.35 Hz (approximately)
Therefore, the frequency of the 3rd harmonic is approximately 75.35 Hz.
Now let's move on to the second part of the question. If the source of the wave is an open organ pipe of the same length as the wire, we can use the equation:
f = nv/(2L)
To find the frequency of the 4th harmonic, we simply need to plug in n = 4 into the equation. The length of the vibrating string and organ pipe is the same, so L remains the same. The velocity of the wave (v) is also the same, as it depends on the properties of the medium through which the wave is traveling.
Using the same values of n and v as in the previous calculation:
f = (4 * v) / (2 * L) = (4 * 49.19 m/s) / (2 * 0.31 m) = 98.38 Hz (approximately)
Therefore, the frequency of the 4th harmonic is approximately 98.38 Hz.