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Spiders may "tune" strands of their webs to give an enhanced response at frequencies corresponding to the frequencies at which desirable prey might struggle. Orb web silk has a typical diameter of 0.0020 mm, and spider silk has a density of 1300 kg/m3. To give a resonance at 200Hz to what tension must a spider adjust a 16cm long strand of silk?

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Answer:

0.0000167283525619 N

Explanation:

[tex]\rho[/tex] = Density of silk = 1300 kg/m³

d = Diameter = 0.002 mm

r = Radius = 0.001 mm

l = Length = 16 cm

f = Frequency = 200 Hz

Mass of the string is

[tex]m=\rho V\\\Rightarrow m=\rho Al\\\Rightarrow m=1300\times \pi (0.001\times 10^{-3})^2\times 0.16\\\Rightarrow m=6.5345127195\times 10^{-10}\ kg[/tex]

Frequency is given by

[tex]f=\dfrac{1}{2l}\sqrt{\dfrac{T}{m/l}}\\\Rightarrow T=4lf^2m\\\Rightarrow T=4\times 0.16\times 200^2\times 6.5345127195\times 10^{-10}\\\Rightarrow T=0.0000167283525619\ N[/tex]

The tension on the string is 0.0000167283525619 N

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