A large ant is standing on the middle of a circus tightrope that is stretched wi

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 A 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. Express the minimum wave amplitude in terms of T_s, mu, lambda, and g.

Explanation / Answer

The ant will feel “weightless” when the only force acting on it is the force of gravity. This
will be true if the rope's acceleration in the downward direction is equal to or greater than
the acceleration of gravity. This would be similar to being in a falling elevator or in orbit
around a body; everything around you accelerates at the same rate and thus appears
“weightless”. However we know that the influence of gravity stretches over infinity at the
speed of light.
Anyway, we're interested in the amplitude of a wave traveling on the rope that
corresponds to a y acceleration of exactly g, since this would be the minimum amplitude.
The acceleration in the y direction can be found by differentiating the y position function
with respect to time twice.
y (x ,t)=Asin(kxwt)
vy=dy/dt = w Acos(kxwt)
ay=dvy/dt =d^2y/dt2 = -w^2 Asin(kxwt)

The maximum downward acceleration occurs when the sine function equals 1. We want
this maximum acceleration to equal that of gravity.
ay(max) = -w^2A=g -> A=g/w^2

We can write this in terms of the tension, linear mass density, wavelength and acceleration
of gravity by revisiting the very first problem of this homework. Recall that
v= w/k
w=vk ,
k=2pi/lamda

,
and that for a rope under tension Ts and with mass per unit length mu
v=(Ts/mu)^1/2

Putting all of this together we see that
A=g/w^2
A=g/(vk)^2
A=gmu/Ts(lamda/2*pi)^2

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