Weird second law of Newton for rotation

[Mad_Eye]

the proof of second law of Newton for rotation goes like this:
take a sphere rotating around some far axis
F_T=ma_T
F_T=m * ALPHA * R
multiple by R

tau = mR² * ALPHA

and we can say it true for every limit mass in a body.
so

sigma tau = sigma (mR²) * ALPHA
sigma tau = I * ALPHA

internal torque=0
thus, sigma tau = sigma external tau

fine, i get it, i think.


but what if we didn't multiple by R?
let's return to here
F_T=m * ALPHA * R

and do not multiple by R

now as far as i see it, this is also true for every limit mass on a body, so
sigma F_T = sigma (mr) * ALPHA
internal forces=0 (?)
thus
sigma external F_T = F_T = sigma (mr) * ALPHA

but that just doest make sense... since it as though it doesn't matter where the force is applied...


thanks a lot for helping...

 

[Gerenuk]

sigma external F_T = F_T = sigma (mr) * ALPHA

but that just doest make sense... since it as though it doesn't matter where the force is applied...

What do you mean by "it doesn't matter..."?
In any case you have to keep in mind that you had vector quantities before. For example one could use a vector \vec{R} which had length R, but direction perpendicular to the displacement and into the direction of the acceleration. Then your equation would be
\sum_i \vec{F_i}=\sum_i (m_i\vec{R}_i(t))\cdot\alpha
which due to the time dependence is not very helpful.

 

[Mad_Eye]

What do you mean by "it doesn't matter..."?
In any case you have to keep in mind that you had vector quantities before. For example one could use a vector \vec{R} which had length R, but direction perpendicular to the displacement and into the direction of the acceleration. Then your equation would be
\sum_i \vec{F_i}=\sum_i (m_i\vec{R}_i(t))\cdot\alpha
which due to the time dependence is not very helpful.

by "it doesn't matters" i simply mean that, wherever the force is exerts relative to the axis, it'll cause the same angular acceleration, which we know not to be true..

and i didn't quite get it... where the time came from?

 

[Gerenuk]

The point is that you need to use exact notation, which encompasses using vectors for the force (otherwise the sum of the forces is not equal to the total external force!).
The time dependence is since the orientation and thus R will change with time.

 

[Mad_Eye]

wow... i still don't get it...
can you write to me the real proof of torque? so i can see how it should be done?
thanks

 

[Gerenuk]

Have a look at
https://www.physicsforums.com/showthread.php?t=363521
It's not the full answer to your question, but feel free to request a special answer :)

There I explain how the general torque law derives for a set of particle.
If the particles form a rigid body, then the proof can be continued. I think about what you would like to hear and some time later I make a post.