Superposition and Interference problem

[habibclan]

I have two questions from the Superposition unit that I don't get it. I'd really any help on these as I have my physics exam on monday! Thanks for any help!

Homework Statement

A steel wire is used to stretch a spring. An oscillating magnetic field drives the steel wire back and forth. A standing wave with three antinodes is created wehn the spring is stretched 8.0 cm. What stretch of spring produces a standing wave with two antinodes?

Here's the picture for it:
http://i196.photobucket.com/albums/aa59/aliatehreem/45.jpg

Homework Equations

λ= 2L/m m=1,2,3 (formula for mth possible wavelength)

The Attempt at a Solution

I really don't know how to tackle this problem, besides the fact that two antinodes means m=2, and so when I use the wavelength formula (assuming that the wavelength remains constant with al lengths of stretching).

For given info, m=3 and L=0.08 m
λ= 2L/m= 2(0.08)/3 = 0.053 m

Therefore, to solve for the length of the second one,
L=λm/2= 0.053*2/2=0.053 m

However, the answer is actually 0.18 m or 18 cm. I don't think I'm understanding this question properly but I just don't understand the way the problem is set up! Can someone help me please.


The second question from the textbook is:

Homework Statement

A 40-cm -long tube has a 40-cm-long insert that can be pulled in and out. A vibrating tuning fork is held next to the tube. As the insert is slowly pulled out, the sound from the tuning fork creates standing waves in the tube when the total length L is 42.5 cm, 56.7 cm, and 70.9 cm. What is the frequency of the tuning fork?

I've drawn the picture here: http://i196.photobucket.com/albums/aa59/aliatehreem/50.jpg

Homework Equations

λ= 4L/m

f= mv/4L

The Attempt at a Solution

Would this be an open-closed tube as the tuning fork blocks one side? I don't know if the wavelength remains constant for all three lengths and so I don't know how to tackle this problem. I would really appreciate if anyone could guide me in the right direction!

 

[tiny-tim]

I have two questions from the Superposition unit that I don't get it. I'd really any help on these as I have my physics exam on monday! Thanks for any help!

Homework Statement

A steel wire is used to stretch a spring. An oscillating magnetic field drives the steel wire back and forth. A standing wave with three antinodes is created wehn the spring is stretched 8.0 cm. What stretch of spring produces a standing wave with two antinodes?

Here's the picture for it:
http://i196.photobucket.com/albums/aa59/aliatehreem/45.jpg

Homework Equations

λ= 2L/m m=1,2,3 (formula for mth possible wavelength)

The Attempt at a Solution

I really don't know how to tackle this problem, besides the fact that two antinodes means m=2, and so when I use the wavelength formula (assuming that the wavelength remains constant with al lengths of stretching).

For given info, m=3 and L=0.08 m
λ= 2L/m= 2(0.08)/3 = 0.053 m

Therefore, to solve for the length of the second one,
L=λm/2= 0.053*2/2=0.053 m

However, the answer is actually 0.18 m or 18 cm. I don't think I'm understanding this question properly but I just don't understand the way the problem is set up! Can someone help me please.

Hi habibclan!

(btw, you don't need to solve the equations completely, you only need the ratios of the lengths! )

You're assuming it's a fixed wavelength, in which case obviously the ratio will be m/m´, in this case 2/3.

(while the correct answer uses the ratio m´²/m²)

But look at what's moving the spring … it's an oscillating magnetic field, so it's the frequency that's fixed.

So the relevant equation is … ?

 

[habibclan]

Hi habibclan!

(btw, you don't need to solve the equations completely, you only need the ratios of the lengths! )

You're assuming it's a fixed wavelength, in which case obviously the ratio will be m/m´, in this case 2/3.

(while the correct answer uses the ratio m´²/m²)

But look at what's moving the spring … it's an oscillating magnetic field, so it's the frequency that's fixed.

So the relevant equation is … ?

You're referring to the frequency equation: f= mv/2L. I tried using that but the answer comes out to 8.33 m, whereas at the back of the textbook, it is 18 cm :S. This is my working:

frequency for the 8 cm one where m=3 as there are 3 antinodes: f= (3 v) / (2*0.08)
frequency for unknown length with m=2: f= (3v) / (2L)

since you're saying frequencies are equal, (3 v) / (2*0.08)= (3v) / (2L), then L=0.053=5.3 cm, however,the correct answer is 18 cm.


What about the second question about the double-slit experiment? Can you please guide me as to how to approach such a question? Thanks a lot =)

 

[tiny-tim]

You're referring to the frequency equation: f= mv/2L.

That's assuming v is constant (then fλ = v).

Do you know an equation that involves the spring constant, k?

 

[habibclan]

That's assuming v is constant (then fλ = v).

Do you know an equation that involves the spring constant, k?

Us= 1/2 k x^2 ?

 

[habibclan]

Can someone help me please?? I've got my exam tomorrow morning!

 

[tiny-tim]

Sorry … forget the spring constant … you can do it simply like this:

tension in a spring is proportional to length;

and wave speed is proortional to square-root of length

(I got this from doing a forum search, https://www.physicsforums.com/archive/index.php/t-152536.html")

But to answer your questions, you don't need to know the spring constant. You just need to know the proportionality between the stretch of the string, to the tension of the spring, to the speed of the wave in the string.

If the frequency is constant, by doubling the stretch of the spring, the tension will double, and the wave speed will increase by 1.41 (square root 2).

In your question, f is constant, and L is proportional to √v.

So … ?