A string is fixed at both ends and vibrating at 120 Hz, which is its third harmonic frequency. The linear density of the string is 4.9x10-3 kg/m, and it is under a tension of 3.6 N. Determine the length of the string.
To determine the length of the string, we need to use the equation for harmonic frequencies of a string:
f = (nv) / (2L)
Where:
f is the frequency of vibration
n is the harmonic number
v is the velocity of the wave traveling through the string
L is the length of the string
Since the frequency given is the third harmonic frequency (n = 3), and the velocity of the wave can be determined using the wave equation:
v = √(T / μ)
Where:
T is the tension in the string
μ is the linear density of the string
Now, we can plug in the given values to find the length of the string.
First, let's find the velocity of the wave:
v = √(T / μ)
= √(3.6 N / 4.9x10^-3 kg/m)
Next, let's calculate the velocity using the given values:
v = √(3.6 N / 4.9x10^-3 kg/m)
≈ √(734.69 m^2/s^2)
≈ 27.09 m/s
Now, we can plug in the values for the frequency (f), harmonic number (n), and velocity (v) into the equation for harmonic frequencies:
f = (nv) / (2L)
120 Hz = (3 * 27.09 m/s) / (2L)
Simplifying the equation:
120 Hz = 81.27 m/s / L
Now, solve for L:
L = 81.27 m/s / 120 Hz
= 0.67725 m
Therefore, the length of the string is approximately 0.67725 meters.
To determine the length of the string, we can use the formula for the frequency of a vibrating string:
f = (n/2L) * sqrt(T/μ)
Where:
- f is the frequency of vibration (120 Hz).
- n is the harmonic number (3rd harmonic in this case).
- L is the length of the string (what we want to find).
- T is the tension in the string (3.6 N).
- μ is the linear mass density of the string (4.9x10^(-3) kg/m).
Rearranging the formula to solve for L:
L = (n/2) * sqrt(T/μ) / f
Substituting the given values:
L = (3/2) * sqrt(3.6 N / 4.9x10^(-3) kg/m) / 120 Hz
Now we can calculate:
L = (3/2) * sqrt(3.6 N / 4.9x10^(-3) kg/m) / 120 Hz
L = (3/2) * sqrt(3.6 N / 4.9x10^(-3) kg/m) / 120 Hz
L ≈ 0.034 m (rounded to three significant figures)
Therefore, the length of the string is approximately 0.034 meters.
This question seems to be missing the radius of the string.
From the equation v = sqrt(T/mu)
we can solve for mu to get: mu = T/V^2
We already have tension, to find velocity we use v = lambda*frequency
Because its a third harmonic frequency the lambda(wavelength) is 2/3 of the string's length.
V= 2/3L(120Hz)
V= 80L
mu = 3.6 N/80L
mu represents mass per unit length
mu = mass/Length
We don't have mass so we substitute density*volume
to get: mu =d*pi*r^2*L/ L
Now we plug this back in to our first equation to solve for L.
d*pi*r^2*L/ L=3.6 N/80L
L=.045/(4.9x10-3*pi*r^2)