Question

A large ant is standing on the middle of a circus tightrope that is stretched with tension T_s. The rope has mass per unit length mu. Wanting to shake the ant off the rope, a tightrope walker moves her foot up and down near the end of the tightrope, generating a sinusoidal transverse wave of wavelength lambda and amplitude A. Assume that the magnitude of the acceleration due to gravity is g.
What is the minimum wave amplitude A_min such that the ant will become momentarily weightless at some point as the wave passes underneath it? Assume that the mass of the ant is too small to have any effect on the wave propagation.

Answer 1

Answer:

$A = \frac{g \lambda^2 \mu}{4\pi^2 T_s}$

Explanation:

As we know that the speed of the wave in string is given as

$v = \sqrt{\frac{T_s}{\mu}}$

now we will have

frequency of the wave is given as

$f = \frac{v}{\lambda}$

$f = \frac{1}{\lambda}\sqrt{\frac{T_s}{\mu}}$

now if the ant will feel weightlessness then we will have

$mg = m\omega^2 A$

so we will have

$A = \frac{g}{\omega^2}$

$A = \frac{g}{4\pi^2 f^2}$

$A = \frac{g \lambda^2 \mu}{4\pi^2 T_s}$