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 3.5 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.

1.Determine the maximum length the string can have.

2.Determine the minimum radius the string can have

3.Determine the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate.

Answer 1

Hi,

Hope you are doing well.

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PART 1:

Given that,

Breaking stress of string,

Maximum tension,

Hence, minimum area of cross-section of the string is given by,

Given that,

Mass of string,

Density of the string,

Therefore, volume of the string is given by,

We know that, volume of a cylindrical string having area of cross section and length is given by,

We have, Volume of the string,

Minimum area of cross-section of the wire,  

Therefore, maximum length of the string is given by,

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PART 2:

Let's take the wire cross-section as a circular cross-section.

We know that, area of a circular cross-section is given by,

We have, Minimum area of cross-section of the wire,  

Therefore, minimum radius the string can have:

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PART 3:

We know that, fundamental frequency of standing waves on this string is given by,

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