What will the wave velocity be in this string?

In summary: I am not sure what the correct solution is or what the problem writer intended. Perhaps there is a typo in the original problem?In summary, the wave velocity of a string with a length of 100 cm and a 10 kg-wt tension, forming 9 nodes when vibrated and becoming symphonic to a 50 Hz tuning fork, is calculated to be 89.534 m/s using the equation V=√(T/m). However, this answer does not match the given frequency of 50 Hz. There appears to be a discrepancy between the given information and the calculated value, as the frequency should be a harmonic of 50 Hz, which would result in a different velocity.
  • #1
Zakiyah Afrin
8
3

Homework Statement

What will be the wave velocity, if a string of 100 cm length is in a 10 kg-wt tension forms 9 nodes when vibrated and becomes symphonic to a 50 Hz tuning fork. Given, Cross sectional area of string, $$A=4.95 mm^2$$ & density, $$d = 0.25 g/cc$$?

Homework Equations

$$ V=\sqrt{\dfrac{T}{m}} $$

m= mass per unit length

T = Tension in string


The Attempt at a Solution


Velocity in a string
$$ V=\sqrt{\dfrac{T}{m}} $$
Where,

Tension in string, $$T= 10\times9.8=98 kN$$
mass per unit length, $$m= \frac{v \times d}{L}=A\times d= 0.012225 kg/m $$
So,$$V = 89.534 m/s$$
.Now the problem is answer should be equal to 50 m/s . Is there any effect of tuning fork on the wave velocity of the string ?
 
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  • #2
Hi Zakiyah Afrin and welcome to PF.

Mass is density multiplied by volume, not density multiplied by cross sectional area. To find the volume, you need the length of the string. How can you find an expression for that?
 
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  • #3
I mean mass per unit length i.e $$m= \frac{v \times d}{L}=A\times d$$
 
  • #4
And what do you get with this correction?

On Edit: Actually, yes m = A d is correct for "mass per unit length". I interpreted "m" as total mass. What do you get for this in units of kilograms per meter?
 
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  • #5
kuruman said:
And what do you get with this correction?
no I calculated in right way. May be there Is something with the frequency
 
  • #6
You appear to have made some mistakes. When you say
##T= 10\times9.8=98 kN## this is not correct. ##T= 10\times9.8=98 N##. ##1~kN=1000~N##. That's minor. You seem to have used 98 N in your calculation so no harm done.
You seem to have made some error in your calculation for ##d##. When you convert 0.25 g/cc, remember that 1 g = 10-3 kg and 1 cc = 10-6 m3.
Now for the important part. You are given the tension ##T##, and enough information to find the linear density ##\mu## (I prefer to use ##m## for total mass). Then it should be a straightforward calculation to find the velocity using ##v=\sqrt{T/\mu}## which is what you attempted. The problem is that if you put in the correct values for ##T## and ##\mu##, you do not get 50 m/s.

However, if you completely ignore the tension and the linear density and assume that "becomes symphonic to a 50 Hz tuning fork" means that the vibrating frequency of the string is a harmonic of 50 Hz (which one?) you can get 50 Hz (how?)

This problem bothers me because the sets of information it provides appear to be internally inconsistent.
 
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  • #7
kuruman said:
And what do you get with this correction?

On Edit: Actually, yes m = A d is correct for "mass per unit length". I interpreted "m" as total mass. What do you get for this in units of kilograms per meter?
as I mentioned 0.01225
 
  • #8
Zakiyah Afrin said:
as I mentioned 0.01225
Repeating the number, does not help if you do not show what went in the calculation. Your number is 10 times what it should be. Also, be sure to include units every time you quote a number. It's a good habit. :smile:
 
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  • #9
Sorry, I did lot of mistakes.
F=98 N
μ=0.001225 kg/m
v= 283.13 m/s
T= 98 N
but frequency for this becomes
141.57 Hz
 
  • #10
Zakiyah Afrin said:
Sorry, I did lot of mistakes.
F=98 N
μ=0.001225 kg/m
v= 283.13 m/s
T= 98 N
but frequency for this becomes
141.57 Hz
Yes, that's what I got when I used ##v=\sqrt{T/\mu}##. Please read post #6. You can find v = 50 m/s if you use v = λ f, but you need to find λ and f separately. Hint: f is not 50 Hz but a harmonic of 50 Hz.
 
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  • #11
kuruman said:
Yes, that's what I got when I used ##v=\sqrt{T/\mu}##. Please read post #6. You can find v = 50 m/s if you use v = λ f, but you need to find λ and f separately. Hint: f is not 50 Hz but a harmonic of 50 Hz.
since, no. of nodes = 9
so it will be 8th harmonic I guess n= 8
f = n f0=8*50=400 Hz
λ = 2*L/n=1/4m
$$v=f λ=400*1/4=100 ms^{-1}$$
but not 50 m\s
 
  • #12
How do you get harmonics from nodes? How many nodes does the fundamental ##n=1## have? Draw a picture if you must.
 
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  • #13
kuruman said:
How do you get harmonics from nodes? How many nodes does the fundamental ##n=1## have? Draw a picture if you must.

upload_2018-10-8_10-6-33.png

for n= 1 number of nodes = 2
 

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  • #14
I have a different counting scheme that excludes the ends, but that's OK as long as you know what you mean. In that case I agree with calculation of 100 m/s in post #11. Unless I misunderstand what "becomes symphonic" means, I don't see how one gets 50 m/s.
 
  • #15
Then, there is a question fault right ?
 
  • #16
Yes. Whether the v = 100 m/s or 50 m/s it is still incompatible with the tension and the linear density.
 

Related to What will the wave velocity be in this string?

1. What factors affect the wave velocity in a string?

The wave velocity in a string is affected by tension and density of the string material. These two factors are directly proportional to the wave velocity, meaning that as tension or density increases, the wave velocity also increases.

2. How can I calculate the wave velocity in a string?

The wave velocity in a string can be calculated using the equation v = √(T/μ), where v is the wave velocity, T is the tension in the string, and μ is the density of the string material.

3. What units is the wave velocity measured in?

The wave velocity in a string is typically measured in meters per second (m/s).

4. Can the wave velocity in a string be greater than the speed of light?

No, the wave velocity in a string cannot be greater than the speed of light. The speed of light is the maximum possible speed for any object, including waves in a string.

5. Does the wave velocity in a string change depending on the amplitude of the wave?

No, the wave velocity in a string is independent of the amplitude of the wave. The wave velocity is only affected by the tension and density of the string material.

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