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 20 mm and spider silk has a density of 1300 kg/m3
. To have a
fundamental frequency at 100 Hz, to what tension must a spider adjust a 12-cm-long strand
of silk?
To calculate the tension in the spider's silk strand, we can use the equation for the fundamental frequency of a string:
f = (1/2L) * sqrt(T/μ)
Where:
f = fundamental frequency
L = length of the string
T = tension in the string
μ = linear mass density
Given:
Length of the silk strand (L) = 12 cm = 0.12 m
Diameter of the silk strand = 20 mm = 0.02 m
Radius of the silk strand (r) = (0.02 m) / 2 = 0.01 m
Density of spider silk (μ) = 1300 kg/m^3
Fundamental frequency (f) = 100 Hz
We can calculate the linear mass density using the formula:
μ = (m/L) * πr^2
Where:
m = mass of the silk strand
To calculate the mass (m), we can use the formula for the volume of a cylinder:
V = πr^2h
Given:
Length of the silk strand (h) = 0.12 m
Radius of the silk strand (r) = 0.01 m
V = π(0.01 m)^2 * 0.12 m
V ≈ 3.769 x 10^-6 m^3
Since density (μ) = mass/volume, we can rearrange the equation to solve for mass (m):
m = μ * V
m = 1300 kg/m^3 * 3.769 x 10^-6 m^3
m = 4.8937 x 10^-3 kg
Then, we can calculate the linear mass density:
μ = (m/L) * πr^2
μ = (4.8937 x 10^-3 kg / 0.12 m) * π(0.01 m)^2
μ ≈ 0.4267
μ ≈ 0.43 kg/m
Now, we can rearrange the equation for the fundamental frequency to solve for tension (T):
f = (1/2L) * sqrt(T/μ)
T = (f * 2L)^2 * μ
T = (100 Hz * 2 * 0.12 m)^2 * 0.43 kg/m
T ≈ 316.23 N
Therefore, the spider must adjust the tension in the silk strand to approximately 316.23 N in order to have a fundamental frequency of 100 Hz.