Lab Question
For this lab we were asked to determine the Linear Mass Density (μ) of a string, given an oscillator, a pulley and a mass.
Experimental Design
To accomplish this, we attached the string to an oscillator, and hung a mass from the string over a pulley (negligible friction).
Procedure
To determine the μ, we first did some algebra by setting the 2 equations for wavespeed equal to each other:
v = λ*f and v = sqrt(Ft/μ)
μ = (Ft)/(f*λ)^2
So, to solve for μ, we needed to determine the Force of tension, the frequency, and the wavelength. Finding the force of tension is easy; since the system has no acceleration, the tension is equal to the force of gravity acting on the hanging mass. For our experiment, the mass was 0.2 kg, so the force of gravity is that times [approximately] 10. So, the force of tension is 2 N. Finding the frequency and the wavelength is a bit more complicated; we changed the frequency of the string until we found the fundamental frequency (1 antinode in the middle), which was 12 hz. Then we used the wavelength equation: λ = 2L (for the fundamental frequency). Our length was exactly 1 meter, so our λ was a convenient 2 meters.
v = λ*f and v = sqrt(Ft/μ)
μ = (Ft)/(f*λ)^2
So, to solve for μ, we needed to determine the Force of tension, the frequency, and the wavelength. Finding the force of tension is easy; since the system has no acceleration, the tension is equal to the force of gravity acting on the hanging mass. For our experiment, the mass was 0.2 kg, so the force of gravity is that times [approximately] 10. So, the force of tension is 2 N. Finding the frequency and the wavelength is a bit more complicated; we changed the frequency of the string until we found the fundamental frequency (1 antinode in the middle), which was 12 hz. Then we used the wavelength equation: λ = 2L (for the fundamental frequency). Our length was exactly 1 meter, so our λ was a convenient 2 meters.
This leaves us with: Ft = 2 N f = 12hz λ = 2 m μ = (2)/(12*2)^2 μ = 0.00347 To support our results, we did multiple experiments with different frequencies. We experimented with the 2nd, 3rd, and 4th harmonic. The results were the same, since all it did was scale the frequency up as much as it scaled the wavelength down. The changed canceled out to yield the same result. |
Conclusion
Manipulating the different wave equations allows you to determine the linear mass density if you have the force of tension, frequency, and wavelength. Changing the harmonic doesn't affect the μ in any way, since increasing or decreasing the frequency by the scale factor of the harmonic would result in a scale decrease or increase, respectively, of the wavelength. This reinforces the idea that the only way to change the wavespeed is the tension force, since the frequency and the wavelength are directly inversely proportional, and the μ stays the same if the string does.