Help Finding The Tension Force of a String (Not a Homework Problem)

[AFinch]

I'm trying to put together a little physics project, and I've constructed a rudimentary stringed instrument using a 2-liter bottle and some string. The string is fed through the bottom of the bottle and wound around a small dowel at the top end, and a small hole has been cut near middle of the bottle so you can put a finger inside to pluck the string. I want to determine the tension force on the string, and then find the increase in tension with each full turn of the dowel.

I know that the tension can be found from the equation fλ=sqrt(F/u), where f is frequency, F is the tension force and u is the linear density of the string. I've found u by dividing the length of the string by its mass. After winding the string, I use a guitar tuner to find frequencies.

My questions are:

If I use the equation v=λf to find λ, would v simply be the speed of sound in air?

When I wind the string around the dowel, can I even use my original value for u since the string will be stretched? Any suggestions or enlightenment would be much appreciated!

 

[AlephZero]

If I use the equation v=λf to find λ, would v simply be the speed of sound in air?

No. λ is twice the vibrating length of the string. It has nothing to do with the speed of sound in air. The string would vibrate at the same frequency in a vacuum, though you wouldn't be able to hear it.

You can use v=λf to find the "speed of sound" (more accurately, the speed of a traveling wave) in the string. That will not be the same as the speed of sound in air, and as your other formula shows, it depends on the tension in the string and its mass per unit length.

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html