Standing Waves (tension vs. wavelength)

[furiouspoodle]

Homework Statement

I'm trying to decode a graph of Tension vs. $$\lambda^2$$ of standing waves on a string to understand the actual meaning of the slope. I also need to derive the equation for the line from two expressions:

$$v= \sqrt{T/\mu}$$
$$v= f\lambda$$

Homework Equations

$$\lambda=2L/n$$
$$\mu = m/L$$

The Attempt at a Solution

I'm starting to think I'm completely missing something in my initial approach.

$$v = \sqrt {T/\mu}$$

$$v^2 = T/\mu$$

$$v = f\lambda$$

$$v^2 = (f\lambda)^2$$

$$T/\mu = f^2\lambda^2$$

$$T = \mu f^2\lambda^2$$

$$T = \frac {\mu f^2 4L^2}{n^2}$$

I feel like it should lead to a clearer solution than this one, but I'm not sure what. I first thought that the slope was simply a surface tension (dynes/cm^2) on the string but I don't see how that relates to the equation or slope I'm trying to find.

I also tried this:

$$f = \frac {v}{\lambda}$$

$$f = \frac {\sqrt {T/\mu}}{\lambda}$$

$$f^2 = \frac {T/\mu}{\lambda^2}$$

$$\mu f^2 = \frac {T}{\lambda^2}$$

which seems closer, but I don't know how to apply it in a usable manner. Any help would be appreciated.

 

[TSny]

The OP found the correct relation $T = (\mu f^2)\lambda^2$. Thus, the physical interpretation of the slope of the $T$ vs. $\lambda^2$ graph is that it represents the product of $\mu$ and $f^2$. I think that should be an acceptable answer.

 

[kuruman]

I grappled with this question without coming to a satisfactory conclusion. The OP mentions a graph of tension vs. λ² which presumably was produced from collected data. If one varies the tension and finds the frequency of the fundamental for fixed wavelength λ = 2L, the relation between the tension and the frequency is $T=4L^2\mu f^2.$ In that case, a plot of T vs. (2Lf)² would yield a straight line with slope equal to $\mu.$ We need additional information about how this graph was produced from what data. It might elucidate the need for plotting tension vs. wavelength squared.

 

[TSny]

I grappled with this question without coming to a satisfactory conclusion. The OP mentions a graph of tension vs. λ² which presumably was produced from collected data. If one varies the tension and finds the frequency of the fundamental for fixed wavelength λ = 2L, the relation between the tension and the frequency is $T=4L^2\mu f^2.$ In that case, a plot of T vs. (2Lf)² would yield a straight line with slope equal to $\mu.$ We need additional information about how this graph was produced from what data. It might elucidate the need for plotting tension vs. wavelength squared.

Ok. Thanks.

Maybe it's just me, but if the graph plots T vs λ² as stated by the OP, then it doesn't make sense to me to think of the wavelength as fixed.

 

[kuruman]

Ok. Thanks.

Maybe it's just me, but if the graph plots T vs λ² as stated by the OP, then it doesn't make sense to me to think of the wavelength as fixed.

It doesn't make much sense to me either. In a standing wave experiment the wavelength can be varied by varying the driving frequency at fixed tension. I can think of an experiment in which one fixes the driving frequency, varies the tension and then finds the separation between the ends that will produce the same order harmonic. Then one can make sense of a T vs. λ² plot.

 

[TSny]

I can think of an experiment in which one fixes the driving frequency, varies the tension...

Yes, that's what I had in mind. The tension can be due to some hanging weights. The weights are varied to get different standing waves (with different wavelengths) with the same fixed frequency.

 

[kuruman]

Yes, that's what I had in mind. The tension can be due to some hanging weights. The weights are varied to get different standing waves (with different wavelengths) with the same fixed frequency.

With near-zero probability of getting clarifications from the OP we can only guess $\dots$