# Weird second law of Newton for rotation

In summary, according to the proof of the second law of Newton, the torque on a rotating object is equal to the sum of the external torques on each of its parts.
the proof of second law of Newton for rotation goes like this:
take a sphere rotating around some far axis
FT=maT
FT=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?
FT=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 FT = sigma (mr) * ALPHA
internal forces=0 (?)
thus
sigma external FT = FT = 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...

sigma external FT = FT = 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.

Gerenuk said:
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?

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.

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

Have a look at
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.

## 1. What is the weird second law of Newton for rotation?

The weird second law of Newton for rotation, also known as the parallel axis theorem, states that the moment of inertia of a body rotating about an axis is equal to the sum of the moment of inertia of the body's center of mass and the moment of inertia of the body's mass distributed at a distance from the axis.

## 2. How is the weird second law of Newton for rotation different from the regular second law of motion?

The regular second law of motion, also known as the law of inertia, states that an object will remain at rest or in uniform motion unless acted upon by an external force. The weird second law of Newton for rotation is specific to rotational motion and describes how the distribution of mass affects the moment of inertia of a rotating body.

## 3. Why is it called the "weird" second law of Newton for rotation?

The term "weird" is often used to describe this law because it is not as well-known or commonly taught as the regular second law of motion. It also has a more complex and specific application compared to the regular second law, which can make it seem strange or unusual to some people.

## 4. How is the weird second law of Newton for rotation used in real-world applications?

The weird second law of Newton for rotation is used in various engineering and physics applications, such as designing and analyzing rotating machinery, calculating the stability of structures, and predicting the behavior of rotating objects in space. It is also used in sports, such as figure skating and gymnastics, to understand and improve the performance of athletes.

## 5. Can the weird second law of Newton for rotation be derived from other laws of motion?

Yes, the weird second law of Newton for rotation can be derived from the regular second law of motion and the definition of moment of inertia. By understanding the relationship between these laws and concepts, one can derive the parallel axis theorem and apply it to solve problems involving rotational motion.

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