Resonance and rules for mathematical equations

In summary, the wavelength of the lowest note that can resonate within an air column 42 cm in length and closed at both ends is \frac{2l}{n}. To get to this equation, you need to multiply both sides of the given equation (l = \frac{n\lambda}{2}) by 2, then divide by n, and switch the sides. If you need a refresher on algebra, there are many online resources available such as themathpage.com, sosmath.com, and coolmath.com.
  • #1
Spookie71
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Homework Statement



What is the wavelength of the lowest note that can resonate within an air column 42 cm in length and closed at both ends.

Homework Equations



Given: l = 42 cm
n = 1

Required: [tex]\lambda[/tex]

Analysis: l = [tex]\frac{n\lambda}{2}[/tex]

therefore [tex]\lambda[/tex] = [tex]\frac{2l}{n}[/tex]

I don't know how this came to be, could someone please explain.

I've just gone back to high school after 15 years and would appreciate if you could also forward me on to some good links for rules for mathematical equations such as these.

Thanks
Scott
 
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  • #2
The steps to go from:

[tex]l = \frac{n\lambda}{2}[/tex] to [tex]\lambda = \frac{2l}{n}[/tex] ?

start at:

[tex]l = \frac{n\lambda}{2}[/tex]

first multiply both sides by 2.

that gives:

[tex]2l = 2\times \frac{n\lambda}{2}[/tex]

on the right side, the 2 in the numerator cancels with the 2 in the denominator, so

[tex]2l = n\lambda[/tex]

Then divide both sides by n.

[tex]\frac{2l}{n} = \frac{n\lambda}{n}[/tex]

on the right side, the n in the numerator cancels with the n in the denominaotr. so,

[tex]\frac{2l}{n} = \lambda[/tex]

Then just switch sides.

[tex]\lambda = \frac{2l}{n}[/tex]
 
  • #3
Thanks Learningphysics for helping

Scott
 
  • #4

1. What is resonance and how does it relate to mathematical equations?

Resonance is a phenomenon in which an object or system vibrates at its natural frequency when exposed to an external force. In mathematical equations, resonance occurs when the solution to an equation matches the natural frequency of a system, resulting in a large amplitude response.

2. Can resonance occur in any type of mathematical equation?

Yes, resonance can occur in various types of mathematical equations, including linear and nonlinear equations. It is not limited to a specific type of equation but depends on the natural frequency of the system and the external force applied.

3. What are the rules for solving equations involving resonance?

The rules for solving equations involving resonance vary depending on the type of equation and the system. However, some general rules include identifying the natural frequency of the system, determining the external force, and finding the solution that matches the natural frequency to achieve resonance.

4. How can resonance be beneficial in scientific research?

Resonance can be beneficial in scientific research as it allows scientists to study the properties and behaviors of systems at their natural frequencies. This can provide valuable insights into how different systems work and how they respond to external forces, which can be applied in various fields such as engineering, physics, and biology.

5. Are there any potential dangers of resonance in mathematical equations?

In certain situations, resonance can lead to unwanted and potentially dangerous effects. For example, in structural engineering, resonance can cause buildings and bridges to vibrate, leading to structural damage. It is important to understand and control resonance in order to prevent such hazards.

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