Missing sketch in K&K Mechanics book

[Seydlitz]

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.

 

[jtbell]

The diagram is indeed missing.

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 $\Delta \vec A \rightarrow 0$.

 

[Seydlitz]

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?

 

[jtbell]

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.

 

[Seydlitz]

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. :)