(II) A small fly of mass 0.28 g is caught in a spider’s web. The web oscillates predominantly with a frequency of 4.0 Hz. At what frequency would you expect the web to oscillate if an insect of mass 0.46 g is trapped?

Verified step by step guidance

1. Step 1: Recognize that the problem involves oscillations and the relationship between mass and frequency. The frequency of oscillation is inversely proportional to the square root of the mass attached to the web. This relationship can be expressed as \( f \propto \frac{1}{\sqrt{m}} \).
2. Step 2: Write the formula for the frequency ratio between two different masses. Let \( f_1 \) and \( f_2 \) be the frequencies corresponding to masses \( m_1 \) and \( m_2 \), respectively. The relationship is given by \( \frac{f_2}{f_1} = \sqrt{\frac{m_1}{m_2}} \).
3. Step 3: Substitute the given values into the formula. Here, \( f_1 = 4.0 \, \text{Hz} \), \( m_1 = 0.28 \, \text{g} \), and \( m_2 = 0.46 \, \text{g} \). Ensure the masses are in the same units (grams in this case).
4. Step 4: Rearrange the formula to solve for \( f_2 \), the new frequency: \( f_2 = f_1 \cdot \sqrt{\frac{m_1}{m_2}} \).
5. Step 5: Perform the substitution and simplify the expression to find the new frequency. Note that you do not need to calculate the final numerical value here, but the process involves substituting \( f_1 = 4.0 \), \( m_1 = 0.28 \), and \( m_2 = 0.46 \) into the formula.

Key Concepts

Oscillation Frequency

Oscillation frequency refers to the number of complete cycles a system undergoes in a unit of time, typically measured in Hertz (Hz). In the context of a mass-spring system, the frequency is influenced by the mass of the object and the stiffness of the spring. A heavier mass generally results in a lower frequency of oscillation, as it takes more time to complete each cycle.

Mass-Spring System

A mass-spring system is a common model in physics that describes how a mass attached to a spring behaves when displaced from its equilibrium position. The system's oscillation frequency is determined by the mass of the object and the spring constant, which measures the spring's stiffness. The relationship is given by the formula f = (1/2π)√(k/m), where f is the frequency, k is the spring constant, and m is the mass.

Damping and Energy Loss

Damping refers to the gradual loss of amplitude in oscillating systems due to energy dissipation, often caused by friction or air resistance. While the question focuses on frequency, understanding damping is important as it can affect the oscillation characteristics of the web. In a real-world scenario, the spider's web may experience some damping, but for the purpose of calculating frequency, we often assume ideal conditions.