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Andre45 [30]
2 years ago
8

A string or rope will break apart if it is placed under too much tensile stress. Thicker ropes can withstand more tension withou

t breaking because the thicker the rope, the greater the cross-sectional area and the smaller the stress. One type of steel has density 7890 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 4.0 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 to determine (a) the maximum length and minimum radius the string can have; (b) the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate.
Business
1 answer:
kifflom [539]2 years ago
4 0

Answer:

a. radius = 0.0006m = 0.6mm and length =0.393m = 393mm

b. frequency =377.86Hz

Explanation:

Given:

mass of steel= 4g = 0.004kg

density of steel = 7890kg/m3

tensile stress of steel 7.0x10⁸

tension load =900N

from the density, we will calculate for the Volume of the steel string

density = mass/volume

volume = mass/density = 0.004/7890 = 5.07 x 10⁻⁷ m³

from the tensile stress will can get the maximum base Area of the string ,

tensile stress = load/area =

7x10⁸ = 900/A

A = 900/7x10⁸ = 1.29x10⁻⁶ m²

Area = πr²

area/pi = 4.105x10-7

radius = 0.0006m = 0.6mm

volume = Area x length

length = vol/area =(5.07 x 10⁻⁷ m³)/1.29x10⁻⁶ m² = 0.393m

b. the highest possible frequency is given by:

F = \frac{\sqrt{\frac{T}{m/L} }}{2L}

where T= tension, m=mass, L=length

F = \frac{\sqrt{\frac{900}{0.004kg/0.393m} }}{2*0.393m}

= 297.04/0.786

frequency =377.86Hz

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