Wave Propagation in a Hanging Rope: Speed at Top vs Bottom

In summary, the tension in the rope and the connection to the ceiling both play a role in the speed of a wave traveling up a long rope. While the tension may seem small, it greatly affects the speed of the wave. In ideal situations, the wave will not lose energy as it moves up, but in reality, it will always lose some energy.
  • #1

Homework Statement


If a wave is started up a long rope hanging from the ceiling, it will not climb at constant velocity. Why? Will the wave be traveling faster or slower at the top than the bottom? (Hint: Is the tension of the rope a factor?)



Homework Equations


v=square root (F/u) (tension over mass per unit length) ?


The Attempt at a Solution



I don't know how to do this.
 
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  • #2
"I don't know how to do this" is not an attempt.
We can't help you unless you try something, or put some thoughts out there.
 
  • #3
Ok...well I considered that the tension in the rope may be greater at the top because of the weight of the rope below it, but it would have to be a very large rope for that to make much difference. I also considered that if the rope is connected to the ceiling that may reduce the ability of the rope to carry a wave which would slow the wave down.
 
  • #4
Is it possible for a wave to lose energy as it moves up?
 
  • #5
Awesome, you've got it all right there.

The rope is connected to the ceiling, but oddly enough, it does NOT slow the wave down.. even though it seems like it should. The ceiling connection will "dampen" the wave (decrease its amplitude/strength) but not slow it down.

You're also very correct about the tension... and although the effects of that tension change seem small, think about this:
How much tension is there at the very very bottom of the rope, like the very last 1/10,000th of an inch (where there is essentially nothing below it)?
How does that compare to the top of the rope?

You've totally got the answer.
 
  • #6
forsberg21 said:
Is it possible for a wave to lose energy as it moves up?

In an ideal situation, no it cannot lose energy.
In actuality it will always lose some energy.

When i said that it loses amplitude as it goes up - its still conserving energy, it just takes more energy (more work) to create smaller amplitude changes as you get closer to where its fixed to the ceiling.
 
  • #7
sweet thank you
 

1. What is wave propagation in a hanging rope?

Wave propagation in a hanging rope refers to the movement of energy or disturbance along the length of the rope. This can occur when the rope is disturbed at one end, causing a wave to travel through the rope to the other end.

2. How does wave propagation in a hanging rope differ from other materials?

The wave propagation in a hanging rope is different from other materials because the rope is able to move freely and is not constrained by a solid surface. This allows for a more complex wave motion as the rope can bend and twist in addition to moving up and down.

3. What factors affect the speed of a wave in a hanging rope?

The speed of a wave in a hanging rope is affected by the tension, mass, and length of the rope. A higher tension, lower mass, and shorter length will result in a faster wave speed, while a lower tension, higher mass, and longer length will result in a slower wave speed.

4. Why does the wave speed differ at the top and bottom of a hanging rope?

The wave speed differs at the top and bottom of a hanging rope because of the varying tension levels. The tension is higher at the top of the rope and decreases towards the bottom, resulting in a faster wave speed at the top and a slower wave speed at the bottom.

5. How can we calculate the wave speed at the top and bottom of a hanging rope?

The wave speed at the top and bottom of a hanging rope can be calculated by using the formula v = √(T/μ), where v is the wave speed, T is the tension, and μ is the mass per unit length of the rope. This calculation will give the wave speed at any point along the rope, including the top and bottom.

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