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 μ. 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 λ and amplitude a. assume that the magnitude of the free-fall acceleration is g.

Answer 1

The minimum wave amplitude in terms of Ts, μ, λ, and g is;

A = λ²gμ/(4π²Ts)

Since the giant ant is said to be standing on a stretched rope, the ant's positional motion would be represented by a sinusoidal wave:

y = A sin(ωt)

The ant is said to lose its sense of weight when its acceleration equals that of gravity. In order to obtain the acceleration, let's first determine the second derivative of the position equation.

y' = Aω cos(ωt)

y" = -Aω² sin(ωt)

Acceleration; a = -Aω² sin(ωt)

Now, the maximum acceleration will occur when sin(ωt) = 1 .  This will happen when a = -g.  

Thus;

-g = -Aω²

A = g/ω²

        Formula for angular frequency in terms of velocity and wavelength is;

ω = 2πv/ λ

Thus;

A = g/(2πv/λ)²

A = λ²g/(2λπ)²v²

       Now, the formula for velocity of the wave in terms of tension Ts and linear density μ is:

v = √(Ts/μ)

Thus;

v² = Ts/μ

Thus;

A = λ²g/((2π)²(Ts/μ))

Learn more about sinusoidal transverse here

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