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 7720 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 4.5 g of this type of steel. In use, the guitar string must be able to withstand a tension of 900 N without breaking.

A) Determine the maximum length the string can have.

B)  Determine the minimum radius the string can have.

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

Explanation / Answer

volume of string = mass / density

= (4.5 x 10^-3 kg) / (7720 kg/m^3 )

V = 5.829 x 10^-7 m^3

stress = F / A

7 x 10^8 = 900 / A

A = 1.286 x 10^-6 m^2

and V = A L

L = 0.45 m .........Ans(A)

(B) A = pi r^2

1.286 x 10^-6 = pi r^2

r = 6.40 x 10^-4 m

(C) for fundamental frequency,

wavelength = 2 L = 0.90 m

linear mass density = mass / length = 4.5 x 10^-3 / 0.45

= 0.01 kg / m

v = sqrt[ T / linear mass density ]

v = 300 m/s

f = v / wavelength = 333.33 Hz ..........Ans

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