# Wave velocity in a free-hanging rope

• CharliH
In summary, to find the time for a transverse wave to travel the length of a uniform rope hanging freely from the ceiling, we can use the equation v = \sqrt{\tau/\mu} and set up axes parallel to the x-axis. By letting m(x) represent the mass of the rope below x and using the equation m(x) = \mu x and \tau (x) = m(x)g, we can find the velocity as a function of time, v(x) = \sqrt{gx}. Then, by integrating this equation and setting the limits from x = 0 to x = L, we can obtain the required equation t0 = 2\sqrt{L/g}.
CharliH

## Homework Statement

A uniform rope of length L hangs freely from the ceiling. Show that the time for a transverse wave to travel the length of the rope is t0 = $$2\sqrt{L/g}$$.

## Homework Equations

v = $$\sqrt{\tau/\mu}$$. (Where $$\tau$$ is the tension and $$\mu$$ the linear density of the rope.)

## The Attempt at a Solution

Set up axes so that the rope is parallel to the x-axis, with the bottom of the rope at the origin.

Let m(x) represent the mass of the rope below x. Then $$m(x) = \mu x$$
giving $$\tau (x) = m(x)g = \mu g x$$
so $$v (x) = \sqrt{\mu g x/\mu} = \sqrt{gx}$$

Also $$L = \int^{t_0}_{0} vdt$$

I can see that velocity is a function of time and that integrating will give me something at least similar to the required equation, but I can't figure out how to get v in terms of t. Or maybe I should be getting x in terms of t. I couldn't find that either, though.

v = dx/dt = sqrt(gx)

dx/sqrt(gx) = dt.

Now find the integration and put the limits x = 0 to x = L

Ohhh, I get it now. Thanks!

## 1. What is wave velocity in a free-hanging rope?

The wave velocity in a free-hanging rope refers to the speed at which a wave travels through the rope when it is in a state of equilibrium or free movement.

## 2. How is wave velocity in a free-hanging rope calculated?

The wave velocity in a free-hanging rope can be calculated by dividing the tension in the rope by the linear density of the rope, taking the square root of this value, and then multiplying it by the square root of the force constant of the rope material. This can also be written as v = √(T/μ) * √(k), where v is the wave velocity, T is the tension, μ is the linear density, and k is the force constant.

## 3. What factors can affect the wave velocity in a free-hanging rope?

The wave velocity in a free-hanging rope can be affected by various factors such as the tension, linear density, and force constant of the rope material. Other factors that can influence the wave velocity include the length and thickness of the rope, as well as the temperature and humidity of the surrounding environment.

## 4. How does the wave velocity in a free-hanging rope relate to frequency and wavelength?

The wave velocity in a free-hanging rope is directly proportional to the frequency and wavelength of the wave. This means that as the frequency or wavelength increases, the wave velocity also increases, and vice versa.

## 5. Can the wave velocity in a free-hanging rope be changed?

Yes, the wave velocity in a free-hanging rope can be changed by altering the factors that affect it, such as adjusting the tension or changing the material of the rope. Additionally, the wave velocity can also be modified by adding weights or changing the length or thickness of the rope.

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