Waves in a vertical hanging string

In summary: Anyone see how to get that into a known closed form?"There doesn't seem to be a closed form for it, but there is a formula for the speed as a function of the length and the mass. It's given by the following equation:v = sqrt(F_T/M)
  • #1
Summer95
36
0
I want to make sure I have this right before I move on to the rest of this problem!

1. Homework Statement


A string with mas M and length L is hanging freely and the mass is uniformly distributed along its length.

Homework Equations



a: Fend the tension in the string as a function of x, M, L , and g

b: Show that speed of transverse wave v is x dependent and find v as a function of x, M, L, and g.

c:

The Attempt at a Solution



a: ##F_{T}=\frac{L-x}{L}Mg##

b: ##v=\sqrt{\frac{F_{T}}{\mu }}=\sqrt{\frac{\frac{L-x}{L}Mg}{\frac{M}{L}}}=\sqrt{\frac{L(L-x)g}{L}}=\sqrt{(L-x)g}##
 
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  • #2
Summer95 said:
a: ##F_{T}=\frac{L-x}{L}Mg##

b: ##v=\sqrt{\frac{F_{T}}{\mu }}=\sqrt{\frac{\frac{L-x}{L}Mg}{\frac{M}{L}}}=\sqrt{\frac{L(L-x)g}{L}}=\sqrt{(L-x)g}##
Looks right. (Assuming x is measured from the top :)
 
  • #3
It's an interesting question to think about. It seems to be saying that the speed goes to zero at the bottom end of the string. Does that mean that a wave in the string will stop dead at the bottom end of the string? Does that seem to be what really would happen in a vertically hanging string?
 
  • #4
DEvens said:
It's an interesting question to think about. It seems to be saying that the speed goes to zero at the bottom end of the string. Does that mean that a wave in the string will stop dead at the bottom end of the string? Does that seem to be what really would happen in a vertically hanging string?
Thinking about this some more, I'm not comfortable with simply treating it as a normal traveling wave with a speed that changes along the length. Writing out the differential equation and applying separation of variables, the time-dependent part is SHM, but for the x-dependent (after normalising somewhat) I get xX"+X'+X=0. That has the series solution ##\Sigma \frac{x^i}{{i!}^2}##. Anyone see how to get that into a known closed form?
 
  • #5


As a scientist, your solution for the tension and speed of transverse waves in a vertical hanging string is correct. However, there are a few things to consider and clarify:

1. The equation for tension in the string as a function of x, M, L, and g should be ##F_{T}=\frac{x}{L}Mg##. This is because the weight of the string is evenly distributed along its length, so the tension at any point x will be proportional to the distance from the top of the string (x) divided by the total length of the string (L).

2. The equation for the speed of transverse waves in the string is correct, but it is important to note that this is the speed at any point x along the string. The actual speed of the wave will vary depending on the location along the string.

3. It would be helpful to define the variables in your equations, such as specifying that ##F_{T}## is the tension in the string, ##x## is the distance from the top of the string, ##M## is the mass of the string, ##L## is the length of the string, and ##g## is the acceleration due to gravity.

Overall, your understanding and approach to the problem is correct. Just make sure to clarify your equations and variables for a more comprehensive solution.
 

1. What causes waves to form in a vertical hanging string?

Waves in a vertical hanging string are caused by a disturbance or force acting on the string. This can be from a hand, an object hitting the string, or even a vibration from sound waves.

2. What types of waves can be seen in a vertical hanging string?

The two main types of waves that can be seen in a vertical hanging string are transverse waves and longitudinal waves. Transverse waves have a perpendicular motion to the direction of the wave, while longitudinal waves have a parallel motion to the direction of the wave.

3. How do the properties of a string affect the waves formed?

The properties of a string, such as tension, length, and density, affect the speed and frequency of the waves formed. Higher tension and shorter length result in faster waves, while higher density results in slower waves.

4. What is the relationship between wavelength and frequency in a vertical hanging string?

The wavelength and frequency of waves in a vertical hanging string are inversely proportional. This means that as the frequency increases, the wavelength decreases, and vice versa.

5. How can the speed of waves in a vertical hanging string be calculated?

The speed of waves in a vertical hanging string can be calculated using the formula v = √(T/μ), where v is the speed, T is the tension, and μ is the mass per unit length of the string.

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