Question

A string or rope will break apart if it is placed under too much tensile stress. Thicker ropes can withstand more tension without breaking because the thicker the rope, the greater the cross-sectional area and the smaller the stress. One type of steel has density 7850 kg/m3 and will break if the tensile stress exceeds 7.0×108N/m2. You want to make a guitar string from a mass of 3.8 g of this type of steel. In use, the guitar string must be able to withstand a tension of 900 N without breaking. Your job is the following.

a)Determine the maximum length the string can have. Express your answer using two significant figures.

b) Determine the minimum radius the string can have. Express your answer using two significant figures.

c) Determine the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate. Express your answer using two significant figures.   

This question is from MateringPhysics.

Answer 1

a). There are much more complicated ways out there, but here is what I did: You know that density=mass/volume. Solve for volume: V=mass/density=.0038/7850=4.84*10^-11 You also know that stress=force/area. Solve for area: A=force/stress. Your value for tensile stress is missing from your question but it is in the problem. V=m^3 and A=m^2, so V/A= the length of the string b)A=pi*r^2. You know the area, solve for r. r=sqrt(A/pi). c)frequency=(velocity)/(lambda). velocity=sqrt(tension/u). u=mass/length. Putting these equations together results in: f=sqrt(T*L/m)/lambda. Fundamental frequency means that lambda=2*L So: f=sqrt(T*L/m)/(2L) All of these variables are known, plug in and solve.