Accelerationg of rotating mass *along* the axis of rotation

[birulami]

Says Wikipedia: "The moment of inertia is a measure of an object's resistance to any change in its state of rotation".

Now consider a rotating mass $m$ that I would like to accelerate along its axis of rotation by $a$. Does this count as a "change in its state of motion"? Will it resist the acceleration more that just $F=m\times a$. And if yes, how much?

Thanks,
Harald.

 

[Simon Bridge]

Says Wikipedia: "The moment of inertia is a measure of an object's resistance to any change in its state of rotation".

Now consider a rotating mass $m$ that I would like to accelerate along its axis of rotation by $a$. Does this count as a "change in its state of motion"?

yes. Newton's laws.

Will it resist the acceleration more that just $F=m\times a$.

no. this is a linear acceleration - regular inertia is all you need.

Lets be sure I understand you: something is freely rotating about its center of mass - the rotation takes place in the x-y plane so the angular momentum points in the +z direction ... the z axis is the axis of rotation.

To accelerate the object in the +z direction, you apply an unbalanced force in the z direction through the center of mass. a_z=F_z/m is correct.

An arbitrary force applied to a free body will have a component through the center of mass giving rise to a linear acceleration by F_r=ma and another perpendicular to that giving rise to an angular acceleration by rF_t=Iα

 

[birulami]

Yes, that was what I was after. The $m\times a$ should have been $m\cdot a$. And yes, the force should point at the center of mass as to not tilt the axis of rotation.

Thanks,
Harald.

 

[Simon Bridge]

Since m is a scalar, and a is a vector, it should be just $m\vec{a}$ ... don't worry about it ;)

I could have said that, for an arbitrary force F at position vector r from the center of mass, then $\vec{r}\wedge\vec{F}=I\vec{\alpha}$ and $\vec{r}\cdot\vec{F} = m\vec{a}$

You realize that the Earth is a rotating body being accelerated by an unbalanced force acting through it's center of mass?

Anyway, knowing how a general vector works on a rigid body should help you now.

 

[PJC01]

I would respond by saying that yes, accelerating a rotating mass along its axis of rotation does count as a change in its state of motion. This is because the mass is already in motion due to its rotation, and accelerating it along its axis of rotation would result in a change in its velocity and therefore its state of motion.

The amount of resistance to this acceleration would depend on the moment of inertia of the rotating mass. The moment of inertia takes into account the mass distribution and shape of the object, so a higher moment of inertia would mean a greater resistance to changes in its state of motion.

In terms of the equation F=ma, this would still apply in calculating the force needed to accelerate the rotating mass along its axis. However, the moment of inertia would also need to be taken into account in order to fully understand the resistance to this acceleration.

In summary, the moment of inertia is an important factor in understanding the resistance to changes in the state of motion of a rotating mass, and it would be necessary to consider in determining the force needed to accelerate the mass along its axis of rotation.