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 Ts. The rope has mass per unit length . 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 and amplitude A. Assume that the magnitude of the acceleration due to gravity is g.

Q

What is the minimum wave amplitude Amin 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 Ts, , , and g.

Explanation / Answer

wavelength is given
velocity = sqrt(tension / density)
frequency= velocity/wavelength
angular freq (omega) = 2 pi frequency

So far, so easy. Once we have the frequency, we can calculate the motion of the ant.

ant position = amplitude * sin ((omega)t)
ant velocity = omega amplitude cos ((omega)t)
ant accel = - (omega)^2 amplitude sin ((omega)t)
I got the velocity and accel by differentiating position and velocity with respect to t.

Max ant accel = (omega)^2 * amplitude

Set that equal to g (to make the ant weightless, the acceleration has to be the same as gravity's), and solve for the amplitude
amp = g / (omega)^2

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