A string or rope will break apart if it is placed under too much tensile stress.

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 7730 and will break if the tensile stress exceeds . You want to make a guitar string from a mass of 3.7 of this type of steel. In use, the guitar string must be able to withstand a tension of 900 without breaking. (a) Determine the maximum length the string can have.

Explanation / Answer

This problem requires that you just put a lot of little piecestogether. First, if you know the max tension is 900 N, find theminimum cross sectional area:     Max stress = Max tension / min area or   area = tension / stress = 900 N / 7.0 x 108 N/m2 = 1.2857 x10-6 m2 Also, you can determine the volume of the wire becauseyou have the mass and density:     mass = density * volume  so       V = m / = 4x 10-3 kg / 7800 kg / m3 = 0.5128 x10-6 m3 And... the volume must be the cross sectional area timeslength, so     area * length =volume    or    L = V / A = 0.5128 x10-6 m3 / 1.2857 x 10-6m2 = 0.39886 m = 0.399m     minimum radius?   Well, use thefact that the area is r2 and    r2 = A / =1.2857 x 10-6 m2 / = 0.4093 x 10-6 m2   so    r = 0.640 mm Highest fundamental freq goes with the longest length andis     f = v /2L       where v = ( tension / )1/2 So    v = ( 900 / (0.004 / 0.399) )1/2 = 299.6 m/s And    f = 299.6 / 2 * 0.399  = 375 Hz And... the volume must be the cross sectional area timeslength, so     area * length =volume    or    L = V / A = 0.5128 x10-6 m3 / 1.2857 x 10-6m2 = 0.39886 m = 0.399m     minimum radius?   Well, use thefact that the area is r2 and    r2 = A / =1.2857 x 10-6 m2 / = 0.4093 x 10-6 m2   so    r = 0.640 mm Highest fundamental freq goes with the longest length andis     f = v /2L       where v = ( tension / )1/2 So    v = ( 900 / (0.004 / 0.399) )1/2 = 299.6 m/s And    f = 299.6 / 2 * 0.399  = 375 Hz

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