Question

An ant with mass m kg is standing peacefully on top of a horizontal stretched rope. The rope has mass per unit length 2.88 kg/m and is under tension 35 N. Without warning your nemesis Throckmorton starts a sinusoidal transverse wave of wavelength 0.85 m propagating along the rope. The motion of the rope is in the vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that the mass of the ant is so small that its presence has no effect on the propagation of the wave.

Answer 1

Concepts and reason

The concepts used to solve this problem are speed of the wave, amplitude of a wave, angular frequency, and linear mass density.

First, use the expressions for wave speed and frequency to find the angular frequency.

Then, use the expression for maximum acceleration to find the amplitude of the wave.

Fundamentals

The expression for the maximum acceleration is given below:

Here, the amplitude of oscillation is and the maximum acceleration is .

The expression that relates angular frequency and linear frequency is as follows:

The expression for the speed of wave is given below:

Here, the speed of wave is .

The expression for the speed of a wave along a stretched string is given below:

Here, the speed of the wave is , the tension on string is , and the linear mass density is .

The expression for the speed of a wave along a stretched string is given below:

Substitute for and for .

The expression for the speed of wave is given below:

Substitute for and for .

The expression that relates angular frequency and linear frequency is as follows:

Substitute for and for .

The expression for the maximum vertical acceleration is given below:

Rearrange,

Here, the acceleration is the acceleration due to gravity. Hence, the expression becomes

Substitute for and for .

Ans:

The maximum amplitude of oscillation is .