Transverse Waves- Verticle Rope

[bollocks748]

Homework Statement

A heavy rope 3 m long is attached to the ceiling and allowed to hang freely. Let y = 0 denote the bottom end of the rope. To get started on this problem, imagine cutting the rope at an arbitrary value of y. Draw a free body diagram of each of the two pieces of rope to determine the tension at the point where the rope was cut.

3.1 Determine the propagation speed of transverse waves on the rope and show that this speed is independent of the rope's mass and overall length.


3.2 How long would the rope have to be in order for the maximum propagation speed to be equal to the speed of sound in air (which we will take to be 330 m/s)?


3.3 Calculate the time it takes for a transverse wave to travel from the bottom of the 3 m long rope to the top and then back to the bottom.


3.4 Compare this round-trip time to that for a horizontal rope with the same tension as the average tension of the vertical rope.



2. Homework Equations and

The Attempt at a Solution

All right, here's what I tried:

3.1 If you cut the rope at an arbitrary point y, you can get the tension by the gravitational force on that section of the rope, which would be y*g*mu, with mu being the mass per unit length of the rope. Since propagation speed is sqr(tension/mu), it cancels out to sqrt(y*g), not being dependent on the mass or the total length of the rope.

3.2 This one was easy, I just set 330 m/s to sqrt(y*g), and found y to be 1.1x10^4 m.

3.3 This is where I get stuck. All of the wave equations I know don't let me simplify this enough to be able to solve for T. I don't know the wavelength,amplitude or frequency, so how can I solve for the period?

I really appreciate any help! Thanks in advance ).

 

[bollocks748]

Just didn't want to get bumped off. :-)

 

[thomate1]

Hello,

It seems like you have a good understanding of the concepts involved in this problem. To answer your question about finding the time it takes for a transverse wave to travel from the bottom of the rope to the top and back, we need to use the formula for the period of a wave, which is T = 1/f. In this case, we can calculate the frequency by using the equation v = fλ, where v is the propagation speed and λ is the wavelength. Since we know the propagation speed (from 3.1) and we can calculate the wavelength by using the length of the rope, we can then solve for the frequency. Once we have the frequency, we can plug it into the formula for the period to find the time it takes for the wave to travel from the bottom to the top and back.

As for the comparison in 3.4, we can use the same equations to calculate the period for the horizontal rope and compare it to the period calculated in 3.3.

I hope this helps! Let me know if you have any further questions.

Best,