Question

A large ant is standing on the middle of acircus tightrope that is stretched with tension Ts. The rope has mass per unit length μ. Wanting to shake the ant off the rope, a tightropewalker moves her foot up and down near the end of the tightrope,generating a sinusoidal transverse wave of wavelength γ and amplitude A. Assume that the magnitude of the acceleration due to gravity is g. What is the minimum waveamplitude Amin such that the ant will become momentarily weightless atsome point as the wave passes underneath it? Assume that the massof the ant is too small to have any effect on the wavepropagation. Express the minimum wave amplitude interms of Ts, μ, γ, and g.

Answer 1

Answer:

Explanation:

The ant , at some point of time will be standing on the crest of moving wave passing under its feet . At that position the downward acceleration of the ant will be equal to its weight . Hence it will feel weightlessness . So at that time

Acceleration = g

Acceleration at the top position = ω²A where ω is angular velocity and A is amplitude of vibration ( transverse ) of crest .

ω =2 π n where n is frequency of transverse vibration of particles. This angular frequency will be equal to angular  frequency of wave travelling on the rope .

frequency  of travelling  wave

n.= $\frac{1}{\gamma}\sqrt{\frac{T}{\mu} }$

ω =2 π  x $\frac{1}{\gamma}\sqrt{\frac{T}{\mu}$

ω² = 4π² x T/γ²μ}

For weightlessness

ω² A = g

A= g x γ²μ / 4π² T