What Amplitude Makes an Ant Weightless on a Vibrating Rope?

[jimbo71]

Homework Statement

An ant with mass "m" is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length "mu" and is under tension "F" .
Without warning, Throckmorton starts a sinusoidal transverse wave of wavelength "lambda" propagating along the rope. The motion of the rope is in a vertical plane.

What minimum wave amplitude will make the ant become momentarily weightless? Assume that "m" is so small that the presence of the ant has no effect on the propagation of the wave.
Express your answer in terms of the variables "m","mu","lambda","F" and appropriate

Homework Equations

lambda=2pi/frequency
k=omega/velocity
v=(F/mu)^1/2

The Attempt at a Solution

I have no idea where to start this problem. What must be true for the ant to be weightless? Please help

 

[Doc Al]

Hint: Consider the acceleration of the rope at the ant's location.

 

[minger]

This is a tricky problem. Imagine a wave propagating. The wave has a wavelength which describes the space between waves, amplitude which describes the height of the wave, and a phase angle which describes the location of the wave:
$$z(x,t) = A\sin(\omega t + \phi)$$
What are looking for is the wavelength or period needed to lift the ant up. We know that the ant will feel weightless when there is a vertical force equivalent to its own mass. How to convert displacement to force? We know that force equals mass times acceleration, so if we can get an acceleration on the ant, we can get a force.

To get the acceleration of the ant, we differentiate the displacement function. Let's assume the ant lies at x=0, and ignore the phase angle for now. The velocity is then:
$$v(x,t) = A\omega\sin(\omega t)$$
Immediately we see that the period of the wave is certainly an influencing factor in the solution.

This should get you started. Let us know if you need further help.

 

[jimbo71]

so a(x,t)=-k^2Acos(kx-wt)=-1? because Fant and mant are equal?

 

[jimbo71]

i need more help with this problem

 

[minger]

Under what condition will the ant be "weightless"? Think in terms of of airplane pilots; they are "weightless" when they experience zero g's, right? What does this mean?

 

[Doc Al]

Imagine you are in an elevator standing on a scale. If the elevator isn't accelerating, the scale will read your normal weight. Under what conditions will the scale read zero?