Question

when an insect ventures onto a spiderweb, a slight vibration is set up, alerting the spider.
the density of spider silk is approximately 1.3 g/cm³, and its diameter varies considerably depending on the type of spider, but 3.0 mm is typical.
if the web is under a tension of 0.47 n when a small beetle crawls onto it 33 cm from the spider, how long will it take for the spider to receive the first waves from the beetle?
express your answer to two significant figures in milliseconds.

Explanation:

Step1: Calculate the linear mass density

The formula for linear mass density \(\mu=
ho A\), where \(
ho = 1.3\space g/cm^{3}=1300\space kg/m^{3}\), and \(A=\pi r^{2}\). Given \(d = 3.0\space mm=3\times10^{- 3}\space m\), so \(r = 1.5\times10^{-3}\space m\). Then \(A=\pi(1.5\times 10^{-3})^{2}=7.0686\times10^{-6}\space m^{2}\). So \(\mu=
ho A=1300\times7.0686\times10^{-6}=9.19\times10^{-3}\space kg/m\)

Step2: Calculate the wave speed

The wave speed on a string is \(v=\sqrt{\frac{T}{\mu}}\), where \(T = 0.47\space N\) and \(\mu = 9.19\times10^{-3}\space kg/m\). So \(v=\sqrt{\frac{0.47}{9.19\times10^{-3}}}=\sqrt{51.14}\approx7.15\space m/s\)

Step3: Calculate the time

The distance \(x = 33\space cm = 0.33\space m\). Using the formula \(t=\frac{x}{v}\), so \(t=\frac{0.33}{7.15}\approx0.046\space s = 46\space ms\)

Answer:

\(46\space ms\)