A small fly of mass 0.26 is caught in a spider's web. The web vibrates predominately with a frequency of 4.6 .
a)What is the value of the effective spring stiffness constant for the web?
b.)At what frequency would you expect the web to vibrate if an insect of mass 0.52 were trapped?
2PI f= sqrt (k/m)
solve for k.
To calculate the effective spring stiffness constant of the web, we can use the formula:
Effective Spring Stiffness Constant (k) = (4π^2) * (mass) * (frequency^2)
a) Let's calculate the value of the effective spring stiffness constant for the web:
Given:
Mass of the fly (m) = 0.26 g = 0.26 * 10^-3 kg
Frequency of vibration (f) = 4.6 Hz
Using the formula:
k = (4 * π^2) * (m) * (f^2)
k = (4 * 3.1416^2) * (0.26 * 10^-3) * (4.6^2)
k ≈ 0.058 N/m
Therefore, the value of the effective spring stiffness constant for the web is approximately 0.058 N/m.
b) Now, to find the expected frequency of vibration if an insect of mass 0.52 g is trapped:
Given:
Mass of the insect (m) = 0.52 g = 0.52 * 10^-3 kg
We can use the formula:
Frequency of vibration (f) = (1/2π) * sqrt(k/m)
Using the given value of the effective spring stiffness constant for the web (k ≈ 0.058 N/m) and the mass of the insect (0.52 * 10^-3 kg), we can solve for the frequency:
f = (1/2 * 3.1416) * sqrt(0.058 / (0.52 * 10^-3))
f ≈ 1.30 Hz
Therefore, if an insect of mass 0.52 g is trapped, we would expect the web to vibrate at a frequency of approximately 1.30 Hz.