Tension in a Guitar String and Frequency

PhysicsAmateur

I have a lab to find the relationship between the two. I know the relationship, but I have to be able to measure the force of tension in a guitar string, or calculate it WITHOUT using the frequency. Is there an equation to solve for string tension where is it set up like a guitar string that I am missing?

If not, if I were to pin a guitar string so that it remains stationary, set it over the pulley, add a weight to the end hanging off of the pulley and add a mass, would the mass provide me the tension as well? Finally, if I were to use this, I would have difficulty getting the string tight enough to produce a frequency. Is there a workaround for this?

Please help, this is been stumping me for the past week.

sophiecentaur

Hanging a large mass on the end of the string will provide any tension you want. The string can obviously support it because that is the tension in normal operation. A mass (several kg) on a pulley is a lot cheaper than using any form of force meter for that sort of tension. It's a fairly common experiment. The way I have always excited the string has been to pass a current through the string and to put a fat magnet right next to it near an antinode. It achieves a very light coupling which doesn't affect the measurement.

As the formula involves three variables, you have to know two of them in order to calculate the third - (frequency).

You need to clamp the whole thing firmly or the mass will pull the whole thing over.

PhysicsAmateur

Thanks for the reply! I decided to do your idea with the pulley and as of right now, I have a plank of wood lying flat on a desk and a makeshift pole attatched to the plank. One end of the string is fastened securely onto the pole, and rather than using a pulley, could I just lay the string over another block of wood(where a weight will be attached to dangle over a desk)? Basically, must it be a pulley on the other end of the string? Or can it be substituted with another block of wood.

Then that way I could calculate the force of tension using fG=fT=ma?

Rap

The tension in the string will be the force from the mass minus the friction force as it goes around the corner. A piece of wood has a lot of friction, a pulley has much less, so definitely use a pulley. If you could just hang the weight from the string while it was vertical, there would be zero friction - that would be the best, but then you couldn't make the length exactly what you want it to be.

sophiecentaur

A guitar has a nut and a bridge at either end of the string. These are of some hard, low friction material and have a groove filed into them to define the length of the string vibration. If you use a nut (fret) at the end with the hanging mass, you will reduce any friction effect. Also, lift the vibrating length of the string and let it go, to allow things to settle - guitarists do this when tuning, to equalise the tension on the string and the tension at the tuning head.
Your 'pole' needs to be pretty strong (guitar neck strength) or it may flex and contribute to the frequency of the vibrations. Why not just use the 'plank', which sounds stronger. How do you intend to excite the oscillations?
Hanging the string vertically would be a possible solution but you would then need to clamp it both ends to keep in contact with the nut and bridge. I would go for the horizontal option as you can then get at everything and have somewhere for your measuring setup. If you haven't a pulley, a round bar would do.

LastOneStanding

Additional tip: once you have everything set up, pick a frequency in the typical range of that guitar string and solve for the required tension. Calculate the dangling mass required to produce this tension. That way you have a rough idea of the size of the masses you will need to be dangling—otherwise, you might pick a bunch that all give frequencies too high or too low for convenient measurement.

Edit: but then don't use something the exact mass you calculated since already know what frequency that's supposed to give...

PhysicsAmateur

Thanks for the help guys. I decided to use a plank and pulley. One end of the string is nailed to the plank(through the plastic end of the string), and the other end is laid over a pulley which will be hanging over a table to prevent friction. Then from there I would simply pluck the string either by hand or with a pick to excite the string and produce a frequency.

Again, using this set-up, assuming there is no friction between the string and pulley, would the equation to find the tension of the string be F=mg, where m is the mass of the weight used and g as acceleration of gravity?

Finally, I just have one worry, and that is when I pluck the string, the pulley will move and no sound will be produced. On a piece of wood, the string would actually dig into the wood much like a bridge on the guitar. If that doesn't happen since I am using a pulley will it still produce a sound?(I'll be using 5 pounds and 10 pounds as my weights if that helps).

LastOneStanding

Yes.
Just having the pulley lightly touch the string forces the vibration to have a node at the point. The part of the string in contact with the pulley will move very little.

Chestermiller

If you want to measure the tension in a guitar string, apply a known force sideways at the center of the string. The observed sideways displacement divided by half the string length will be equal to the sideways force divided by twice the string tension.

LastOneStanding

Except that the string needs to be bent an appreciable amount to get any sort of precision, and bending the string changes the thing you're trying to measure. Unless you're suggesting doing it to the string attached to the pulley, in which case why bother?

Chestermiller

You can pull sideways with a smooth lubricated hook that has a much larger cross sectional diameter than the guitar string. This will minimize the bending stress.

PhysicsAmateur

Wow, my internet(Rogers) literally went down. All rogers users had no service for the past couple of hours. Thanks for all of the responses guys! It really helped. I'm going to be completing the lab in a couple days and I'll respond back how it goes.

Cheers!

sophiecentaur

I am still concerned about your frequency measurement. It is far easier to see the steady state resonance from a sinusoidal input than to analyse the frequencies produced by plucking. All that's required is a lab signal generator with a low impedance output and a strong magnet (and perhaps an accurate frequency counter, if the sig gen doesn't have an accurate display)

sophiecentaur

The problem with using the pulley to define the 'end' of the string is that the curve of the pulley will modify the effective length of the string whilst it is oscillating (end effect is a common problem). This can be avoided with a 'knife edge' nut at each end of the string. Aamof, the vibration of the string itself will help to settle any differences between the tensions on the vibrating portion and the tensioning portion.

Rap

Good point.

sophiecentaur

Measuring the frequency is still an important issue - whilst we're discussing the accuracy of measuring the tension, which is hardly even a problem in comparison. Frequency counters often have quite a problem with harmonic-rich waveforms.