Are the Red Terms Equal to Zero in This Cross Product Problem?

[sparkle123]

Background: we're trying to show that the rate of change of angular momentum of an object about its center of mass (position given by R) is equal to the total torque about R.

Why are the terms in red equal to 0? If anything, shouldn't the terms circled in in blue be equal to zero since the vectors R and a_i are parallel and r_i` and a<sub>i</sub>` are parallel?

Thank you!

 

[tms]

What would it mean if, as you say, the acceleration of each individual particle were parallel to $\vec R$?

 

[sparkle123]

Then there is no angular momentum and only linear momentum?

EDIT: actually i made an error with my question. the new image is attached.
If r_i` and a<sub>i</sub>` are parallel, shouldn't the cross-product be 0? So in the last line, the Ʃ m_ir_i` × A would be left instead?

 

[tms]

Then there is no angular momentum and only linear momentum?

No. Reexamine your assumption.

EDIT: actually i made an error with my question. the new image is attached.
If r_i` and a<sub>i</sub>` are parallel, shouldn't the cross-product be 0?

Why do you think they are parallel?

 

[sparkle123]

No. Reexamine your assumption.Why do you think they are parallel?

Isn't a the double derivative of r (so they are parallel)?
Thanks again!

 

[sparkle123]

^bump

 

[tms]

That doesn't mean they must be parallel. If it did, it would mean that you could change the forces on an object just by moving the origin of the coordinate system.