To determine the time it takes for a vibration to reach a spider on its web, one calculates the wave speed using tension and the silk's linear mass density, then divides the distance by this speed. The result is given in milliseconds.
To find out how long it takes for the spider to detect the vibrations caused by the beetle, we need to calculate the wave speed on the web using the tension and the density. The speed of a wave on a string (or spider silk) can be determined using the formula v = √(T/μ), where v is the wave speed, T is the tension, and μ is the linear mass density of the silk. Given the tension T = 0.50 N, and the diameter of the silk d = 3.0 mm, we can calculate the linear density μ given the density of spider silk ρ = 1.3 g/cm³.
First, determine the cross-sectional area A = π(d/2)^2 and then the linear mass density μ using the formula μ = A × ρ. Calculate the speed v and use it to find the time t = distance/v, with the distance being 39 cm between the beetle and the spider.
After calculating the speed of the vibration on the web, we divide the distance from beetle to spider by this speed to find the time it takes for the vibration to reach the spider. The final answer, in milliseconds, is expressed to two significant figures.