Derivation of parallel axis theorem- understanding integration with a dot product.

Question

I am considering this proof of the parallel axis theorem (note that I have recently posted a question about this, but I now have different queries that I feel would be better suited to a new post):

So I am confused about how they got from the second from the last line to the penultimate line (yes, I have read the comment below the derivation- several times!)

In the question I asked about this previously, I did query how they got from one line to the next in a comment (it was this post here) and I kindly received a reply that "the definition of I0I0 becomes ∫(r2−(n⋅r)2)dm∫(r2−(n⋅r)2)dm, which deals with the terms with two rrs, and linearity of the integral gives ∫a⋅rdm=a⋅∫rdm∫a⋅rdm=a⋅∫rdm, which is what happens to the linear terms"

However I am very much confused by this. Specifically,

• Why is it that the $\textbf a$ can be taken out of the integral even though it is DOTTED WITH THE $\textbf r$. I can see that the same 'trick' was used for integrating the $2(\textbf n . \textbf r)(\textbf n.\textbf a)$ term. Are there specific cases when you can/can't do this? I appreciate that this might be a rather large topic, so I would be very grateful if someone could even just provide me with a link that deals with this well.

Answer 1

I don't know what the notation used in the book is exactly, but if you suppose $\mathbb{a}=(a_{1},a_{2},a_{3})$ and $\mathbb{r}=(r_{1},r_{2},r_{3})$ then you can see that an expression like $\int \mathbb{a}\cdot \mathbb{r} \,dm$ can be handled so:

$\int \mathbb{a}\cdot \mathbb{r} \,dm= \int (a_{1}r_{1}+a_{2}r_{2}+a_{3}r_{3}) \,dm= a_{1}\int r_{1}\, dm+a_{2}\int r_{2} \, dm+a_{3}\int r_{3} \,dm= \mathbb{a}\cdot \int \mathbb{r}\, dm$

where $\int \mathbb{r}\, dm=\big(\int r_{1}\, dm,\int r_{2}\, dm,\int r_{3}\, dm\big)$, and we use the linearity of the integral.

I hope this answers your question.