Final answer:
The effective spring stiffness constant k for the web with a mass of 0.26 g oscillating at a frequency of 4.2 Hz is approximately 0.59 N/m.
Step-by-step explanation:
To calculate the spring stiffness constant k for a spider's web with a small fly caught in it, which oscillates with a known frequency, we can use the formula for the period T of harmonic motion, T = 2π√(m/k), where m is the mass and k is the spring constant. Since we are given the frequency f, which is the reciprocal of the period (f = 1/T), we can rearrange the formula to solve for k in terms of f and m.
The mass m of the fly is 0.26 g, which we need to convert into kilograms (m = 0.26 g * 10^-3 kg/g = 2.6 * 10^-4 kg). The frequency f of the oscillations is 4.2 Hz. Using the relationship between period and frequency (f = 1/T), and the formula for the period of a mass-spring system (T = 2π√(m/k)), we get the following equations:
f = ¹/₂π√(m/k)
k = (1/(f²)) * (4π² * m)
By substituting the values for m and f:
k = (1/(4.2²)) * (4 * π² * 2.6 * 10^-4 kg)
k ≈ 0.59 N/m
Therefore, the effective spring stiffness constant k for the web is approximately 0.59 N/m.