Wave Speed Dependency on Distance from Top of Hanging Rope

[BryMan92]

Homework Statement

The question asks:
There is a rope, of length l, and of mass m hanging from the ceiling. What is the wavespeed dependency on z, the distance from the top?

Homework Equations

I used:

V=(FL/M)^(1/2)

The Attempt at a Solution

V=(m(Z/L)gl/m)^1/2=sqrt(zg)
The answer is:
SQRT((l-z)g)

I do not know how to attain this answer, and any help would be great! Thanks!

 

[vela]

Uppercase and lowercase letters generally represent different variables. Stick with using one or the other. Don't switch between them randomly.

According to your work, the tension F in the rope is equal to m(z/l)g. That means at the top where z=0, the tension is F(z=0)=0, and at the bottom, where z=l, the tension is F(z=l)=mg. Does that sound right?

 

[BryMan92]

Uppercase and lowercase letters generally represent different variables. Stick with using one or the other. Don't switch between them randomly.

According to your work, the tension F in the rope is equal to m(z/l)g. That means at the top where z=0, the tension is F(z=0)=0, and at the bottom, where z=l, the tension is F(z=l)=mg. Does that sound right?

My thought process was, that if I have, say L = 10, and Z= 3, that would mean, I would have only have 3/10(mg)...but now that I think about it, that actually means I have 7/10(mg) because the tension is holding 7/10 of the rope. Can I think of it as if I hold my finger at a point z, what is below it is the mass of the string? Thank you!

 

[vela]

Imagine cutting the rope at a distance z from the top. Draw the free-body diagram for the bottom part of the rope and then solve for F.

 

[BryMan92]

Imagine cutting the rope at a distance z from the top. Draw the free-body diagram for the bottom part of the rope and then solve for F.

Gotcha, thank you very much for your help! :D