Approximation of string extension length

In summary, the conversation discusses the length extension of a string that is stretched to a height of y when attached to the origin. It is represented by a triangle with length L=\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}. When the displacement y is small, it can be approximated to y^2/2a. This approximation is obtained using Taylor's series. The correct result is actually y^2/2a, not a³.
  • #1
gboff21
50
0

Homework Statement



A string of length a is stretched to a height of y when it is attached to the origin so making a triangle with length L=[itex]\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}[/itex] and therefore a length extension ΔL= [itex]\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a[/itex] which simplifies to [itex]{a(\sqrt{1+\frac{y}{a}^{2}}-1}[\itex].

Apparently when the displacement y is small, it can be approximated to y^2/2a

How does that even work??
 
Physics news on Phys.org
  • #2
Are you sure you got the result right? Seems to me like there should be a³, not a. This works by a method called Taylor's series. You can read about it in wikipedia: http://en.wikipedia.org/wiki/Taylor_series
 
  • #3
Ah of course. I'm familiar with Taylor but I just didn't apply it. However, I've just done it and it is over 2a. Thanks for pointing me down the right track! :)
 

What is "approximation of string extension length"?

"Approximation of string extension length" is a scientific method used to estimate the length of a string that is extended from a known length. It involves using mathematical algorithms and models to calculate the approximate length of a string based on its original length and the factors that affect its extension, such as tension and elasticity.

Why is approximation of string extension length important in science?

Approximation of string extension length is important in science because it allows researchers to make accurate predictions and calculations in a variety of fields. For example, it is used in materials science to determine the strength and flexibility of different materials, and in biology to study the behavior and movement of living organisms.

What factors can affect the accuracy of the approximation of string extension length?

There are several factors that can affect the accuracy of the approximation of string extension length. These include the type of material the string is made of, the amount of tension applied to the string, the angle at which the string is extended, and any external forces acting on the string.

How do scientists use approximation of string extension length in their research?

Scientists use approximation of string extension length in their research by conducting experiments and collecting data on the extension of strings under different conditions. They then analyze the data using mathematical models and algorithms to estimate the length of the extended string and draw conclusions about the properties and behavior of the string.

Are there any limitations to the approximation of string extension length?

Yes, there are limitations to the approximation of string extension length. It is based on mathematical models and assumptions, so it may not be completely accurate in all situations. Additionally, the accuracy of the approximation may be affected by errors in measurements or variations in experimental conditions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
20
Views
452
  • Calculus and Beyond Homework Help
Replies
9
Views
543
  • Calculus and Beyond Homework Help
Replies
1
Views
946
  • Calculus and Beyond Homework Help
Replies
4
Views
831
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
768
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
731
Back
Top