Angular velocity of a loop after being struck by bullet

[Orodruin]

Could you please edit this message? It only shows the plain text, without the implementation of formatting.

No. I have obviously tried. It is reported to staff.

 

[brochesspro]

There is no dot product.

When we expand the vector triple product, we get a linear combination of the vectors inside the cross product whose coefficients are in the form of dot product.

The first term is very relevant.

I know, and that is the very thing I fail to understand.

 

[Orodruin]

No. I have obviously tried. It is reported to staff.

Scrap that, I found the missing }

 

[Orodruin]

I get a vector triple product in the 2nd term.

When we expand the vector triple product,

You don’t need to expand it. It is the definition of $I_r(\vec \omega)$.

Edit: I mean, you can expand it to rewrite it as
$$
m\vec d \times [\vec \omega \times \vec d]
=
md^2 \vec \omega - \vec d (\omega\cdot \vec d) = md^2 \vec \omega
$$
for an expression more familiar in 2D.

 

[brochesspro]

As for the com issue. Consider
$$
\sum_i m_i (\vec x_i - \vec x_r) \times \vec v_r
=
M \left[ \underbrace{\frac 1M \sum_i m_i \vec x_i}_{\equiv \vec x_{cm}} - \vec x_r \frac 1M \underbrace{\sum_i m_i}_{\equiv M} \right] \times \vec v_r
=
(M \vec x_{cm} - M \vec x_r)\times \vec v_r
$$

Why can we take the Xr out of the Σ? Is it not variable? Since Vr represents its rate of change.

You don’t need to expand it. It is the definition of $I_r(\vec \omega)$.

Could you tell its definition?

 

[Orodruin]

Why can we take the Xr out of the Σ? Is it not variable?

The reference point is fixed.

Edit: Even if it were variable it is independent of $i$, which is what the sum is over…

Since Vr represents its rate of change.

No. It represents the velocity of the solid body at $\vec x_r$.

Could you tell its definition?

$$
I_r(\vec \omega) = \sum_i m_i (\vec x_i - \vec x_r)\times [\vec \omega \times (\vec x_i - \vec x_r)]
$$
…

 

[brochesspro]

No. It represents the velocity of the solid body at x→r.

Wait, $\vec x_r$ and $\vec v_r$ are not the position and the velocity of the reference point respectively? Then what do they mean?

 

[Orodruin]

Wait, $\vec x_r$ and $\vec v_r$ are not the position and the velocity of the reference point respectively? Then what do they mean?

Exactly what I said they mean in my last post.