Waves on a String: Mass & Standing Wave Mode

In summary, the conversation discusses the setup of waves on a string, specifically looking at the mass required for the oscillator to set up the fourth harmonic and the standing wave mode for a specific mass. The calculation for the required mass is based on the wavelength, velocity of the wave, and the string tension, while the calculation for the standing wave mode depends on the velocity and frequency of the wave. It is concluded that a specific harmonic number is not necessary for resonance, and the 120Hz wave does not create a standing wave in the string.
  • #1
danago
Gold Member
1,123
4
[SOLVED] Waves On A String

In the figure below, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation L=1.2m, linear density 1.6 g/m, and the oscillator frequency 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if m=1kg?

http://edugen.wiley.com/edugen/courses/crs1650/art/images/halliday8019c16/image_t/tfg044.gif


Part A
For the string to be vibrating in its fourth harmonic, the wavelength of the wave must be half of the string length so that two full waves can fit on the piece of string simultaneously.

[tex]
\lambda = \frac{L}{2} = 0.6m
[/tex]


We can then use this to calculate the velocity of a wave in the string:

[tex]

v = \lambda f = 0.6(120) = 72ms^{ - 1}
[/tex]

Which allows the strings tension to be calculated:

[tex]
v = \sqrt {\frac{T}{\mu }} \Rightarrow T = v^2 \mu = 8.2994N
[/tex]


The mass of the block is therefore 0.85kg.

Part B
Knowing the mass of the block (thus the tension), the velocity of a wave in the string can be calculated:

[tex]
v = \sqrt {\frac{T}{\mu }} = \sqrt {\frac{{9.81}}{{0.0016}}} \approx 78.3ms^{ - 1}
[/tex]



The wavelength of a wave with frequency of 120Hz can then be calculated:

[tex]

\lambda = \frac{v}{f} = \frac{{78.3}}{{120}} = 0.6525m
[/tex]


This wavelength does not corrospond to a specific harmonic number, so would this therefore mean that the 120Hz wave does not create any standing wave in the string?
 
Last edited:
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  • #2
What is Your take on this:
A specific harmonic number is not necessary for resonance.
 
  • #3
physixguru said:
What is Your take on this:
A specific harmonic number is not necessary for resonance.

I guess that's where I am a little confused.

From what i understand, a specific harmonic number is required for resonance.
 
  • #4
danago said:
In the figure below, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. Separation , linear density , and the oscillator frequency . The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. (a) What mass m allows the oscillator to set up the fourth harmonic on the string? (b) What standing wave mode, if any, can be set up if ?
It's difficult to comment on your work since essential information is missing from the problem statement. (When you cut and pasted, some numbers must have gotten lost.)
 
  • #5
Doc Al said:
It's difficult to comment on your work since essential information is missing from the problem statement. (When you cut and pasted, some numbers must have gotten lost.)

Oh sorry, i didnt realize :redface: I've fixed it now.
 
  • #6
danago said:
This wavelength does not corrospond to a specific harmonic number, so would this therefore mean that the 120Hz wave does not create any standing wave in the string?
Sounds right to me.
 
  • #7
Doc Al said:
Sounds right to me.


Ok, good to hear :)

Thanks for the confirmation.
 

1. What is a wave on a string?

A wave on a string is a type of mechanical wave that travels along a taut string or wire. It is created by a disturbance, such as plucking or striking the string, and can propagate in both directions along the string.

2. How does the mass of the string affect the waves?

The mass of the string affects the speed at which the waves travel. A heavier string will have a slower wave speed compared to a lighter string, as the mass of the string affects its tension and stiffness, which in turn affects the speed of the wave.

3. What are standing wave modes?

Standing wave modes are specific patterns that occur when a wave on a string reflects off of an endpoint. These patterns are characterized by points of maximum and minimum amplitude, and are dependent on the length of the string, the frequency of the wave, and the speed of the wave.

4. How is the standing wave mode affected by the mass of the string?

The mass of the string affects the standing wave mode by changing the wave speed, which in turn affects the wavelength and frequency of the wave. A heavier string will have a lower wave speed, resulting in longer wavelengths and lower frequencies for the standing wave modes.

5. Can the standing wave modes be changed by adjusting the tension of the string?

Yes, the standing wave modes can be changed by adjusting the tension of the string. Increasing the tension will increase the wave speed, resulting in shorter wavelengths and higher frequencies for the standing wave modes. Conversely, decreasing the tension will have the opposite effect.

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