Biologists think that some spiders "tune" strands of their web to give enhanced response at frequencies corresponding to those at which desirable prey might struggle. Orb spider web silk has a typical diameter of 20μm, and spider silk has a density of 1300 kg/m³.
To have a fundamental frequency at 150Hz , to what tension must a spider adjust a 14cm -long strand of silk?

Biologists think that some spiders "tune" strands of their web to give enhanced response at frequencies corresponding to those at which desirable prey might struggle. Orb spider web silk has a typical diameter of 20μm, and spider silk has a density of 1300 kg/m³.

To have a fundamental frequency at 150Hz , to what tension must a spider adjust a 14cm -long strand of silk?

l=length of the spider silk, 14cm

velocity of wave = √(T/μ)

where T = tension and

μ = mass per unit length)

λ/2=l

for fundamental frequency λ/2 =14cm

(λ= wavelength of standing wave; as there will be no node

except the endpoints of silk strand)

λ = 28 cm = 0.28 m

and since frequency * wavelength = speed of wave. we have,

150 * 0.28 = √(T/μ) ..................(#)

now μ = mass/length = [volume * density]/length = [(length*area) * density] / length = area * density

= [π * (10 * 10^(-6))²] * 1300 = 13π * 10^(-8).

now putting this in equation (#) we get

150 * 0.28 = √(T/[13π * 10^(-8)]).

thus T = [13π * 10^(-8)] * (42)² =

2.94*10^-10 N/m.