You take a set of measurements for the wavelengths and frequencies of standing waves on a string that is under a tension of 2.20 N. You use your data to create a plot of wavelength (in m) vs inverse frequency (in s) and the linear fit of your graph gives a slope of 19.0. What is the linear mass density of the string in kg/m?
If a standing wave with 10 antinodes occurs at a frequency of 44.1 Hz, at what frequency would you look for the n = 2 standing
To find the linear mass density of the string, we can use the relationship between the linear mass density (µ), tension (T), slope (m), and gravitational acceleration (g):
µ = [ (4 * T) / (m^2 * g) ]
Given:
Tension (T) = 2.20 N
Slope (m) = 19.0
We can substitute these values into the formula to calculate the linear mass density:
µ = [ (4 * 2.20) / (19.0^2 * 9.8) ]
Simplifying the expression:
µ = [ (8.80) / (361 * 9.8) ]
µ ≈ 2.42 x 10^(-5) kg/m
Therefore, the linear mass density of the string is approximately 2.42 x 10^(-5) kg/m.
To find the linear mass density of the string, we can use the relationship between wave speed and linear mass density. The wave speed (v) of a wave traveling on a string is given by the equation:
v = sqrt(T/μ)
Where T is the tension in the string and μ is the linear mass density.
Since we have the tension (T) and the slope of the graph, we can calculate the linear mass density (μ).
First, let's rearrange the equation for the wave speed to solve for μ:
μ = T / v^2
Now, let's find the wave speed (v) using the slope of the graph. The slope of the graph is the ratio of wavelength (λ) to inverse frequency (1/f), so we can write:
slope = λ / (1/f)
Rearranging the equation:
slope = λ * f
Now, we can substitute the values:
slope = 19.0
Next, we need to find the value of wavelength λ and frequency f from the graph. Using the graph, determine the wavelength (x-axis) and inverse frequency (y-axis) values that correspond to the slope value of 19.0.
Let's say the wavelength corresponding to the slope of 19.0 is λ = 0.5 m and the inverse frequency is 1/f = 0.03 s.
Now, we can plug these values into the equation:
slope = λ * f
19.0 = 0.5 * (1/f)
Now, solve for the frequency f:
1/f = 19.0/0.5
1/f = 38.0
f = 1/38.0 = 0.0263 s^(-1)
Now that we have the wavelength λ = 0.5 m and the frequency f = 0.0263 s^(-1), we can substitute these values into the equation for the wave speed:
v = sqrt(T/μ)
We know the tension T is 2.20 N.
So, the equation becomes:
v = sqrt(2.20 / μ)
Now, let's solve for the wave speed v:
v = sqrt(2.20 / μ)
Substituting the wavelength and frequency values:
v = sqrt(2.20 / μ) = 0.5 / 0.0263
Now, square both sides of the equation:
v^2 = (0.5 / 0.0263)^2
v^2 = 18.7745
Now, solve for the wave speed v:
v = sqrt(18.7745) = 4.330 m/s
Finally, we can use the wave speed v and the tension T to find the linear mass density μ:
μ = T / v^2 = 2.20 / (4.330)^2
μ = 0.115 kg/m
Therefore, the linear mass density of the string is 0.115 kg/m.