Physics-Wave concepts/standing waves

[Noles1]

Homework Statement

A string is tied on both ends and under a tension FT. A standing wave is formed on the string. For each of the following statements select the correct option.
1.)If the distance between the two fixed ends of the string is increased, the wavelength that corresponds to the fundamental frequency ...
2.) When the amplitude of the wave is decreased, the fundamental frequency ...
3.) If the distance between the two fixed ends of the string is decreased, the fundamental frequency...
4.) If the distance between the two fixed ends of the string is increased, the wave speed ...
5.) When the tension of the string is increased, the fundamental frequency...
**for each number, the answer (which is the end to the sentence) is either increases, decreases, or stays the same.**

Homework Equations

I'm not really sure what relevant equations would be. I know f=v/wavelength...

The Attempt at a Solution

I've tried a solution with many different combinations of the answer, but I am still unable to figure it out...even after thoroughly reviewing the material in the book/notes.

 

[kuruman]

You need additional relevant equations. Reread the book and see if you can spot them.

1. What equation gives the fundamental frequency for a standing wave on a string?
2. What equation relates the tension and the wave speed?

 

[Noles1]

v=sqrt(tension force/mu) where mu is mass/length. the fundamental frequency would be the wave speed/wavelength...

 

[kuruman]

And how is the wavelength of the fundamental related to the distance between the fixed ends?

 

[Noles1]

wavelength=wave speed/fundamental frequency

 

[kuruman]

wavelength=wave speed/fundamental frequency

Not what I asked. How is the distance in meters between the two fixed ends of the string related to the wavelength in meters. In other words, if the distance between the two fixed ends is L, how is L related to λ?

 

[Noles1]

oh ok, Sorry!

Does L equal the wavelength?

 

[kuruman]

No. Look at the figure in your textbook showing the fundamental standing wave pattern. The fundamental (first harmonic) has zero nodes, the second harmonic has one. Which one represents a full wavelength?

 

[Noles1]

okay, I think I see. The second harmonic is one full wavelength, and the first (fundamental) is half of a wavelength.

 

[Noles1]

would that mean 1,2,3,5 would have the answer stay the same? I'm still not sure...

 

[kuruman]

okay, I think I see. The second harmonic is one full wavelength, and the first (fundamental) is half of a wavelength.

Correct. So you can say that for the fundamental λ = 2L. Now can you answer question 1? If yes, then can you write an expression relating f, v and L?

 

[Noles1]

Would one be stays the same because the fundamental frequency is constant?
Is v/f=2L the expression?

 

[kuruman]

Would one be stays the same because the fundamental frequency is constant?

Not quite. The question is

"If the distance between the two fixed ends of the string is increased, the wavelength that corresponds to the fundamental frequency _________"

You know that

"the distance between the two fixed ends of the string" = L
and that for the fundamental frequency
λ = 2L
so what's the answer and why?

Is v/f=2L the expression?

Yes, but I would rewrite this as

$$f=\frac{v}{2L}$$

Now question 3 asks "If the distance between the two fixed ends of the string is decreased, the fundamental frequency _____" What do you think? What happens to f if L is decreased?

 

[Noles1]

Would it increase because the L is bigger?

Then for #3, it would decrease?

 

[kuruman]

Correct on both accounts. Now before we proceed, let's take stock of what equations we have.

$$v=\lambda f_1 \ \ (1)$$

$$f_1=\frac{v}{2L} \ \ (2)$$

$$v=\sqrt{Tension/\mu} \ \ (3)$$

Now how do you think you should answer questions 2 and 4?

 

[Noles1]

I honestly am not sure about #2. Do the amplitude and the fundamental frequency have any relation?

For #4, would I look at equation 2 and make it v=f*2L?

PS: Thank-you for your help and patience. I am slowly understanding. :)

 

[kuruman]

I honestly am not sure about #2. Do the amplitude and the fundamental frequency have any relation?

There is no amplitude anywhere in the three equations, and that's all there is. So if A is not related to any of the quantities including f, what happens to A if L is decreased?

For #4, would I look at equation 2 and make it v=f*2L?

That relation is certainly true. Now you know that L is increased? What about v? Does it increase, decrease or stay the same? Hint: What does equation (3) say about that?

PS: Thank-you for your help and patience. I am slowly understanding. :)

You are welcome. I am trying to teach you what I call "equation interpretation." Equations do much more than giving you a number when you put in other numbers. They express relationships between quantities.

 

[Noles1]

Since A is not related to any equations involving f, decreasing it would not make a difference to the fundamental frequency, so the f would stay the same, even if L is increased?

and v would decrease with an increase in the length?

 

[kuruman]

Since A is not related to any equations involving f, decreasing it would not make a difference to the fundamental frequency, so the f would stay the same, even if L is increased?

That is correct.

and v would decrease with an increase in the length?

Why do you say that? Which one of the three equations are you invoking? Look at them carefully and think what changes when L changes.

 

[Noles1]

Would the speed stay the same because the frequency is what depends on the length?

 

[kuruman]

Would the speed stay the same because the frequency is what depends on the length?

Yes, the speed will stay the same because equation (3) says that "as long as the tension and μ stay the same, v stays the same." Do you see how it works?

To answer the last question, put equation (3) in equation (2) and interpret what you get in terms of what is being asked. I have to sign off right now, but I will check back in a couple of hours or so.

 

[Noles1]

When I combine equations 2 and 3, it looks to me like an increase in the tension will cause an increase in the fundamental frequency...

ok, sounds good! I will check back later too. Thank-you again!

 

[kuruman]

When I combine equations 2 and 3, it looks to me like an increase in the tension will cause an increase in the fundamental frequency...

That is correct. Think of the application, when you tighten a guitar string the tone (frequency) goes up, does it not?

 

[Noles1]

ok great!

So, 1.) increase 2.)stays the same 3.)decrease 4.)stays the same 5.)increase

Is this correct? Thanks again for helping!

 

[Noles1]

I think I am missing something because I put that in my answer, and it said it was wrong... :(

 

[kuruman]

I think I am missing something because I put that in my answer, and it said it was wrong... :(

You must recheck your answers one by one in terms of the equations. One of them is incorrect. Which one and why?

 

[Noles1]

I am fairly certain that #2, #4, and #5 are right. So would either #1 and #3 be wrong? I'm not really sure even by looking at the equations. I'm sorry! I thought I was understanding better...

 

[kuruman]

I am fairly certain that #2, #4, and #5 are right. So would either #1 and #3 be wrong? I'm not really sure even by looking at the equations. I'm sorry! I thought
I was understanding better...

Give yourself some credit. You are understanding better. You narrowed down the incorrect answer to 1 and 3 and I agree that one of these you answered incorrectly.

In view of what we said so far and the three equations that I put down,

(a) What do you think the answer to 1 should be and why?
(b) What do you think the answer to 3 should be and why?

 

[Noles1]

Well, thank-you! That makes me feel a little better!

ok so...I think #1 should still increase. Would #3 actually increase because if you divide by a smaller length, the frequency would be larger?

 

[Noles1]

I got it! That was right! Thank-you so much for all of your help. You were so helpful, thorough, and you really made me think! Thank-you again!

Is there any way to rate people on the site because I would really like to give you an amazing rating! :)

 

[kuruman]

Is there any way to rate people on the site because I would really like to give you an amazing rating! :)

Thank you for your kind words. Spread the word among your friends that PF rocks. Other than that, you may wish to send a private message to Greg Bernhardt (administrator) and tell him how happy you are. He works hard to keep PF going and he needs to know. There are no ratings. We all do this for free, just for the satisfaction of bringing good understanding of physics to all those who wish to have it.