Chapter 15: Problem 52
An ant with mass m is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length \(\mu\) and is under tension \(F\). Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wavelength \(\lambda\) propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that \(m\) is so small that the presence of the ant has no effect on the propagation of the wave.
Short Answer
Step by step solution
Understand the Condition for Weightlessness
Express the Wave Motion Mathematically
Calculate the Maximum Vertical Acceleration
Relate Maximum Acceleration to Gravity
Find Angular Frequency
Substitute Angular Frequency into the Condition
Express Wave Speed in Terms of Known Quantities
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Wave Motion Equations
- The amplitude \(A\) represents the maximum vertical displacement from the equilibrium position.
- The wavelength \(\lambda\) is the distance over which the wave's shape repeats.
- The angular frequency \(\omega\) determines how fast the wave oscillates over time.
Vertical Acceleration
- At maximum acceleration, the sine part of the equation equals 1, meaning it's pulling down with full strength, which would be equal to \(A \omega^2\).
- When this matches the gravitational acceleration \(g\), any object, like our ant, may become momentarily weightless if on the wave's peak.
Angular Frequency
- Angular frequency helps determine the sinusoidal properties of the wave, dictating how "stretched out" or "condensed" the wave appears over time.
- It is closely related to the periodicity of waves: higher angular frequency indicates faster oscillations.
Wave Speed Calculation
- Wave speed increases with greater tension because tighter ropes allow waves to propagate faster.
- A lower mass per unit length \(\mu\) also results in higher wave speed, as the rope is lighter and easier to move.