91Ó°ÊÓ

In the realm of physics, a mass-spring system is a classic model to understand harmonic motion. This system consists of a mass attached to a spring, capable of moving back and forth. In our exercise, the spider's web behaves like this system. The mass of the tiny fly causes the web to oscillate. These repetitive movements are close to what we call simple harmonic motion.
For a mass-spring system, the ability to return back to its original position after being disturbed depends on the mass and the stiffness of the spring. Understanding this simple yet fascinating interaction helps us solve problems involving oscillations like the movement of the spider’s web.

The spring constant, denoted as \( k \), is a measure of a spring's stiffness or rigidity. It tells us how much force is needed to stretch or compress the spring by a certain distance. In our problem, the web is equivalent to a spring where the spring constant needed to be calculated.
The calculation of the spring constant involved rearranging the frequency formula. By doing so, we used the fly's mass and the given frequency to find the value of \( k \). This resulted in a spring constant of approximately \( 0.079 \, \mathrm{N/m} \), suggesting that the web is quite delicate.

Oscillation frequency refers to how many times the mass on a spring completes a full cycle of movement in one second. In our scenario, this was first calculated for the fly which was trapped in the web. Its frequency was noted to be \( 4.0 \, \mathrm{Hz} \). This means the web oscillated four times every second with the fly caught in it.
We also calculated how the frequency would change if the mass trapped in the web were doubled. By substituting the new mass into the frequency formula, a reduced frequency of \( 2.83 \, \mathrm{Hz} \) was found. This shows that as the mass increases, the frequency of oscillation tends to decrease, meaning the system moves slower with a heavier mass.

Effective spring stiffness refers to the overall ability of a spring to resist deformation. In the context of the spider’s web, it indicates how strongly the web can resist changes in shape when a fly lands on it. This stiffness is quantified through the spring constant \( k \).
Having calculated the spring constant in our exercise, we can see how an increase in mass affects the stiffness perceived by the oscillating system. Even though the physical properties of the web don't change, the effect of the new mass alters how it oscillates. Understanding effective stiffness helps when predicting how different masses will impact oscillatory systems similar to this one.