Final answer:
The effective spring constant k is found using Hooke's Law and is calculated to be 32.7 N/m for the web when a 15-mg insect depresses it by 4.5 mm.
Step-by-step explanation:
To determine the effective spring constant k for a web modeled as a spring, we can use Hooke's Law, which states that the force F exerted by a spring is equal to its spring constant k times the distance x the spring is stretched or compressed. The formula is F = kx. In this scenario, we are dealing with a 15-mg insect that depresses the web by 4.5 mm when it lands on it. Considering that the insect is at rest when the web stops vibrating, the force exerted by the spring is balanced by the weight of the insect, F = mg, where m is the insect's mass and g is the acceleration due to gravity. Converting the given mass to kilograms (1 mg = 1x10-6 kg) and the distance to meters (1 mm = 1x10-3 m), we have:
m = 15x10-6 kg
g = 9.8 m/s2
x = 4.5x10-3 m
Now we can calculate the force F:
F = mg = (15x10-6 kg)(9.8 m/s2) = 1.47x10-4 N
Using the formula F = kx, we can solve for k:
k = F/x = (1.47x10-4 N) / (4.5x10-3 m) = 3.27x101 N/m
Thus, the effective spring constant k is 32.7 N/m.