Missing sketch in K&K Mechanics book

  • Thread starter Seydlitz
  • Start date
  • #1
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.
 

Answers and Replies

  • #2
jtbell
Mentor
15,637
3,683
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##.
 

Attachments

Last edited:
  • #3
263
4
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
jtbell
Mentor
15,637
3,683
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
263
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.

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

Related Threads on Missing sketch in K&K Mechanics book

  • Last Post
Replies
8
Views
832
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
5
Views
603
  • Last Post
Replies
10
Views
1K
Replies
3
Views
3K
  • Last Post
Replies
21
Views
2K
  • Poll
  • Last Post
Replies
7
Views
2K
Replies
2
Views
739
  • Last Post
Replies
2
Views
640
Top