How Does Hartle Derive the Equation from Arc Length to a Trigonometric Integral?

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In summary, Hartle goes from the equation for the arc length to \sqrt{\frac{R^2}{R^2-x^2}}compute dy/dx and plug into 2.10.b. It leads directly to 2.10c. I certainly couldn't see it by eye - but it is a pretty small calculation.compute dy/dx and plug into 2.10.b. It leads directly to 2.10. I certainly couldn't see it by eye - but it is a pretty small calculation.
  • #1
Nano-Passion
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I have no idea how eq 20.10b to eq 20.10c.

Hartle goes from the equation for the arc length to [tex]\sqrt{\frac{R^2}{R^2-x^2}}[/tex]
 

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  • #2
compute dy/dx and plug into 2.10.b. It leads directly to 2.10c. I certainly couldn't see it by eye - but it is a pretty small calculation.
 
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  • #3
PAllen said:
compute dy/dx and plug into 2.10.b. It leads directly to 2.10. I certainly couldn't see it by eye - but it is a pretty small calculation.

Sorry, I'm not able to read between the line here. What do you mean by compute dy/dx? Compute dy/dx of what?
 
  • #4
Nano-Passion said:
Sorry, I'm not able to read between the line here. What do you mean by compute dy/dx? Compute dy/dx of what?

You have x^2+y^2=R^2. Solve for y and take dy/dx. Plug this into where dy/dx is in 2.10.b. You can then simplify to 2.10.c.
 
  • #5
d(x2 + y2) = 2xdx + 2ydy = dR2 = 0

dy/dx = -x/y
 
  • #6
PAllen said:
You have x^2+y^2=R^2. Solve for y and take dy/dx. Plug this into where dy/dx is in 2.10.b. You can then simplify to 2.10.c.

Oh, alright thanks. ^.^
 
  • #7
One more question, do you know how he got -1, 1 as the limits of integration? It looks as if he pulled that number randomly.
 
  • #8
Nano-Passion said:
One more question, do you know how he got -1, 1 as the limits of integration? It looks as if he pulled that number randomly.

A change of integration variable was made such that the new integration variable is dimensionless, i.e., [itex]\xi = x/R[/itex].
 
  • #9
George Jones said:
A change of integration variable was made such that the new integration variable is dimensionless, i.e., [itex]\xi = x/R[/itex].

Okay, I also don't know how that integral will give you Pi. Something is missing in my knowledge-base.
 
  • #10
Nano-Passion said:
Okay, I also don't know how that integral will give you Pi. Something is missing in my knowledge-base.

That integral should be covered in any first course in calculus; or looked up in even the smallest table of integrals; or recognize that it is the circumference of a unit semi-circle. If you're reading this book, you should have calculus book, and can review integration of trigonometric forms.
 
  • #11
PAllen said:
That integral should be covered in any first course in calculus; or looked up in even the smallest table of integrals; or recognize that it is the circumference of a unit semi-circle. If you're reading this book, you should have calculus book, and can review integration of trigonometric forms.

Thank you, I should have realized that. What I also should have done is just evaluated the integral to see that it equals pi.

I suppose I should jump back to classical mechanics, I need to test out of classical mechanics I anyways. I'll come back to this book later.
 

FAQ: How Does Hartle Derive the Equation from Arc Length to a Trigonometric Integral?

What is Simple Derivation and why is it important?

Simple Derivation is a mathematical process of finding the derivative of a function. It is important because it allows us to calculate the instantaneous rate of change of a function at any given point, which has many practical applications in fields such as physics, engineering, and economics.

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To solve a Simple Derivation problem, you need to follow a specific set of steps. First, identify the function and its independent variable. Then, use the power rule, product rule, quotient rule, or chain rule to find the derivative. Finally, simplify the derivative and substitute in the given value to find the instantaneous rate of change.

What are the common mistakes to avoid when doing Simple Derivation?

The most common mistakes in Simple Derivation include forgetting to apply the chain rule, using the wrong rule, not simplifying the derivative, and making mistakes in basic algebraic calculations. It is important to double-check your work and be careful with each step to avoid these mistakes.

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