a 12.5 m long string vibrates as 100 hertz standing wave with nodes at 1.0m and 1.5m from one end of the string and at no points in between the two. Which harmonic is this? What is the string's fundamental frequency?
To determine the harmonic and the fundamental frequency of the vibrating string, we can use the formula:
v = fλ
Where:
v is the velocity of the wave,
f is the frequency of the wave, and
λ is the wavelength of the wave.
Given that the string is 12.5 meters long, the distance between two nodes is 1.5 - 1.0 = 0.5 meters.
Since there are no nodes between the two points, this means that the wavelength of the standing wave is twice the distance between the nodes.
λ = 2 * 0.5
λ = 1 meter
Now, let's calculate the velocity using the formula:
v = fλ
We know that the frequency is 100 hertz.
v = 100 * 1
v = 100 meters/second
Since the velocity of the wave on the string remains constant, we can calculate the fundamental frequency using the formula:
v = f * λ
f = v / λ
f = 100 / 12.5
f = 8 hertz
Therefore, the string's fundamental frequency is 8 hertz.
To determine the harmonic, we can use the formula:
n = (2L / λ) + 1
Where:
n is the harmonic number,
L is the length of the string, and
λ is the wavelength of the wave.
Substituting the given values:
n = (2 * 12.5) / 1 + 1
n = 25 + 1
n = 26
Therefore, this standing wave is the 26th harmonic.
To determine the harmonic and the fundamental frequency of the vibrating string, we need to understand a few concepts related to standing waves.
First, let's define some terms:
- Harmonic: A harmonic is a single frequency component of a complex periodic wave. Harmonics are typically whole number multiples of the fundamental frequency.
- Fundamental Frequency: The fundamental frequency is the lowest frequency component of a complex waveform. It corresponds to the first harmonic.
Now, let's break down the given information:
1. The string length is 12.5 m, and there is a standing wave with nodes at 1.0 m and 1.5 m from one end of the string.
Based on this information, we can determine the distance between the nodes.
Distance between nodes = 1.5 m - 1.0 m = 0.5 m
The distance between nodes represents half a wavelength (λ/2) for the standing wave.
2. We are given that the string vibrates with a frequency of 100 Hz.
Given the frequency (f) and the speed of the wave (v), we can find the wavelength (λ) using the formula:
v = f * λ
Here, we need to determine the wavelength (λ) of the standing wave.
We know that the distance between nodes is half the wavelength. Therefore,
0.5 m = λ/2
Simplifying the equation, we find:
λ = 1 m
Now, we can find the speed of the wave (v) using the formula:
v = λ * f
v = 1 m * 100 Hz
v = 100 m/s
With these values, we can now calculate the fundamental frequency (f1) for the vibrating string using the formula:
f1 = v / λ
f1 = 100 m/s / 12.5 m
f1 = 8 Hz
Therefore, the fundamental frequency of the vibrating string is 8 Hz.
Finally, to determine the harmonic number of the standing wave, we divide the frequency of the standing wave by the fundamental frequency:
Harmonic number = Frequency of standing wave / Fundamental frequency
Harmonic number = 100 Hz / 8 Hz
Harmonic number = 12.5
Therefore, the given standing wave with nodes at 1.0 m and 1.5 m from one end of the string corresponds to the 12.5th harmonic.