Boundary conditions for 2 ropes fixed to a massless ring with a damper

In summary: I get ##R = \frac{Z_1A - Z_2B - bA}{Z_1 + Z_2 +b}## to reduce to ##R = 0## when ##b = Z_1 - Z_2## by taking the derivative of ##T## with respect to ##x##: ##\frac{dT}{dx} = \frac{dZ_1A - dZ_2B - bA}{Z_1 + Z_2 +b}##
  • #1
Redwaves
134
7
Homework Statement
The boundary conditions for 2 ropes fixed to a massless ring with a massless damper
A wave comes from the left (rope 1) to the right (rope 2).
Relevant Equations
##\psi_1 = Ae^{i(\omega t -k_1x)} + Re^{i(\omega t + k_1x)}##
##\psi_2 = Te^{i(\omega t -k_2x)}##
Hi,
I'm not quite sure if I'm correct. I need to find the boundary conditions for 2 ropes ##T_1 \mu_1, T_2 \mu_2## fixed at ##x=0## to a massless ring with a massless damper of force ##F_d - -bv_y##

Here what I think, since the ring and the damper is massless ##\sum F_y = 0##. Thus, ##-T_1 \frac{\partial \psi_1}{\partial x} + T_2 \frac{\partial \psi_2}{\partial x} -b \frac{dy}{dt} = 0##

Then, the height must be the same for the first rope, the ring and the second rope at ##x=0##, thus ##y_d = \psi_1(0,t) = \psi_2(0,t)##
Which mean, A + R = T, since ##Ae^{i(\omega t -k_1x)} + Re^{i(\omega t + k_1x)} = Te^{i(\omega t -k_2x)}## at x=0
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  • #2
Is there a diagram that goes with this problem? I'm having trouble getting all of the terms correct...
 
  • #3
I did a diagram. Is it clear enough?
 
  • #4
Redwaves said:
I did a diagram. Is it clear enough?
Bwaaa. So the two ropes are co-linear, and there is a ring in the middle connecting them with some sort of damper? Is there really no diagram with the problem that you've been given?
 
  • #5
It's exactly like that. I just drew it myself, but it's basically the same. Correct me if I'm wrong, but to find the boundary conditions, I don't think I need this diagram. Since, there are only 3 forces in the y direction and I know that the damper and the ring is massless.

The goal is the find that ##b = Z_1 - Z_2## to have ##R = 0##. However, I found ##R = -Z_1 + Z_2##. I'm pretty sure my boundary conditions is not correct. ##Z_i## are the impedance.
 
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  • #6
##-T_1 \frac{\partial \psi_1}{\partial x} + T_2 \frac{\partial \psi_2}{\partial x} -b \frac{dy}{dt} = 0##
##T = A + R##

With those conditions I get ##R = \frac{Z_1A - Z_2B - bA}{Z_1 + Z_2 +b}##
if ##b = Z_1 - Z_2##, ##R = 0##
This is the correct answer. But are my boundary conditions correct?
 
  • #7
Redwaves said:
##-T_1 \frac{\partial \psi_1}{\partial x} + T_2 \frac{\partial \psi_2}{\partial x} -b \frac{dy}{dt} = 0##
##T = A + R##

With those conditions I get ##R = \frac{Z_1A - Z_2B - bA}{Z_1 + Z_2 +b}##
if ##b = Z_1 - Z_2##, ##R = 0##
This is the correct answer. But are my boundary conditions correct?
This looks good to me, except I think the B in the numerator of R should be A.
 
  • #8
If B in the numerator of R is A, I didn't get 0 if ##b = Z_1 - Z_2##
 
  • #9
Redwaves said:
If B in the numerator of R is A, I didn't get 0 if ##b = Z_1 - Z_2##

Can you show the individual steps of how you get ##R = \frac{Z_1A - Z_2B - bA}{Z_1 + Z_2 +b}## to reduce to ##R = 0## when ##b = Z_1 - Z_2##?
 

Related to Boundary conditions for 2 ropes fixed to a massless ring with a damper

What are boundary conditions for two ropes fixed to a massless ring with a damper?

The boundary conditions for two ropes fixed to a massless ring with a damper are the constraints that dictate how the ropes will behave at the points where they are attached to the ring. These conditions include the length and tension of the ropes, as well as any damping effects caused by the damper.

How do boundary conditions affect the motion of the ropes?

The boundary conditions play a crucial role in determining the motion of the ropes. They determine the amplitude, frequency, and shape of the oscillations of the ropes. The length and tension of the ropes, as well as the damping effect of the damper, all influence the motion of the ropes.

What happens if the boundary conditions are changed?

If the boundary conditions are changed, the motion of the ropes will also change. For example, if the length of the ropes is increased, the amplitude of the oscillations will also increase. Similarly, if the tension in the ropes is increased, the frequency of the oscillations will also increase.

How are the boundary conditions determined?

The boundary conditions for two ropes fixed to a massless ring with a damper can be determined through mathematical equations and physical experiments. The equations take into account the properties of the ropes, the ring, and the damper, while experiments can be used to verify the results and make adjustments if necessary.

Can the boundary conditions be ignored?

No, the boundary conditions cannot be ignored. They are an essential aspect of the system and must be taken into consideration in order to accurately predict the behavior of the ropes. Ignoring the boundary conditions can lead to incorrect results and a misunderstanding of the system's dynamics.

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