Missing sketch in K&K Mechanics book

In summary, the conversation discusses a confusing derivation in the K&K Intro to Mechanics book where the equation ##|\delta A| = 2A \sin{\frac{\delta \theta}{2}}## is mentioned, but the corresponding diagram is missing. It is explained that the equation can be derived by considering one of the right triangles formed by the three vectors in the isosceles triangle. The discrepancy between the diagrams is resolved in the limit as Δt and Δθ approach zero. There is also a discussion on the perpendicularity of ΔA and A, which is necessary for the equation to hold. The conversation ends with the clarification that in the limit, ΔA becomes perpendicular to both A(t) and A(t+
  • #1
Seydlitz
263
4
Hello guys,

I'm currently reading through K&K Intro to Mechanics book, and I'm on page 26 where I encounter a bit odd derivation.

The authors say:
...

Using the angle ##\delta \theta## defined in the sketch,

##|\delta A| = 2A \sin{\frac{\delta \theta}{2}}##
I'm rather lost on this part. I don't know which sketch corresponds to this equation.
 
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  • #2
The diagram is indeed missing.

attachment.php?attachmentid=66201&stc=1&d=1391257399.gif


The three vectors form an isosceles triangle. The dashed line bisects it into two right triangles. To get the equation, consider one of the right triangles.

If you take the diagram at the top left of page 26, and fill in the hypotenuse which is ##\vec A (t + \Delta t)##, you get a similar diagram, but with the right angle in a different location. The discrepancy disappears in the limit as ##\Delta \theta \rightarrow 0## and ##\Delta \vec A \rightarrow 0##.
 

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  • #3
Ok that makes it clear. But it raises another question on why the vector ##\Delta A## is not perpendicular to ##A##? Whereas the book says it is necessary for it to be so because ##A## is not changing in magnitude?
 
  • #4
In the limit as Δt goes to zero (which is what you need to do in order to have the instantaneous derivative), ΔA becomes perpendicular to both A(t) and A(t+Δt). And of course A(t) and A(t+Δt) become equal to each other.
 
  • #5
jtbell said:
In the limit as Δt goes to zero (which is what you need to do in order to have the instantaneous derivative), ΔA becomes perpendicular to both A(t) and A(t+Δt). And of course A(t) and A(t+Δt) become equal to each other.


Ah yes I didn't think of that carefully. Thanks for the remark and the sketch. :)
 

1. What is the missing sketch in the K&K Mechanics book?

The missing sketch in the K&K Mechanics book is an illustration that is supposed to accompany a specific topic or problem in the book. It is a visual aid that can help readers better understand the concept being discussed.

2. Why is the missing sketch important?

The missing sketch is important because it provides a visual representation of the concept being discussed. This can aid in the understanding of the topic and make it easier to grasp for readers. It can also serve as a helpful reference when solving problems.

3. How does the absence of the sketch affect the book?

The absence of the sketch can make it difficult for readers to fully understand the concept being discussed. It may also make it more challenging to solve problems related to that topic. The missing sketch can also cause confusion and frustration for readers.

4. Is there a way to obtain the missing sketch?

If the missing sketch is not included in the book, you may be able to find it online or in other resources. You can also try reaching out to the publisher or the author to request the missing sketch. Additionally, some book editions may have the sketch included, so it may be worth checking if there are different versions available.

5. How can the missing sketch be replaced?

If you are unable to obtain the missing sketch from other sources, you can try creating your own sketch based on the description and information provided in the book. You can also try visualizing the concept in your mind or using other diagrams or illustrations as a substitute. However, it is important to note that the missing sketch may not be accurately replaced, so it is best to try and obtain the original sketch if possible.

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