Questions: A large orb spider has a mass of 0.70 grams. With its silk webbing (mass density = 0.250 g / m ) which has a diameter of 0.0023 millimeters, the spider lowers itself down from a fix point on a tall tree branch, stopping and hanging 1.5 meters below the branch. If a bird lands on the branch above the spider creating a vibration, how long will it take for the spider to feel the vibration? 0.29 s 1.7 s 0.87 s 5.2 s 0.0068 s

Solution Steps

The speed of a wave on a string is given by the formula:

\[ v = \sqrt{\frac{T}{\mu}} \]

where \( T \) is the tension in the string and \( \mu \) is the mass density of the string.

The tension \( T \) in the silk is due to the weight of the spider. The weight \( W \) is given by:

\[ W = mg \]

where \( m = 0.70 \, \text{g} = 0.00070 \, \text{kg} \) is the mass of the spider and \( g = 9.81 \, \text{m/s}^2 \) is the acceleration due to gravity.

\[ T = 0.00070 \times 9.81 = 0.006867 \, \text{N} \]

Substitute the values of \( T \) and \( \mu \) into the wave speed formula:

\[ v = \sqrt{\frac{0.006867}{0.250 \times 10^{-3}}} \]

\[ v = \sqrt{\frac{0.006867}{0.00025}} = \sqrt{27.468} \approx 5.240 \, \text{m/s} \]

The time \( t \) it takes for the vibration to travel the length of the silk (1.5 meters) is given by:

\[ t = \frac{d}{v} \]

where \( d = 1.5 \, \text{m} \).

\[ t = \frac{1.5}{5.240} \approx 0.2869 \, \text{s} \]

Final Answer