What is the relationship between rotating objects and their surfaces?

[tiny-tim]

i thought that the instantaneous axis of rotation can only and always pass through the CM..

no, Doc Al is correct: the instantaneous axis of rotation of a rigid body is defined as the line on which all points are instantaneously stationary

(there always is such a line)

so the centre of mass can lie on it only if it is instantaneously stationary!

(for example, the instantaneous axis of rotation of a rolling wheel is the "horizontal" line through the point of contact, which of course goes nowhere near the centre of mass!)

or did you mean, if there are no external forces (not even gravity)? even in that case, you'd still need to choose a frame of reference in which the centre of mass was stationary!

 

[fisico30]

Ok, thanks.
For some reason I always thought it was the CM the pivotal point in the rolling wheel example but it is the contact point instead...thanks.

Every point inside an extended object has 3 principal axes of rotation associated to it. For instance, point A has 3 orthogonal principal axes. Point B also have its 3 principal axes (with different eigenvalues than the eigenvalues of point B). The triad for point A can be oriented differently than the triad for point B, correct?
Is the triad of principal axes passing through the center of mass the one that has the smallest eigenvalues among all sets of eigenvalues, hence the smallest moments of inertia?

Torque is force times lever arm times sine of the angle. For the same force, why, physically an conceptually speaking, does a longer lever arm increase the ability to cause a change in rotation? Why does a force that is applied farther from the pivot point cause a larger effect? Has this phenomenon been discovered empirically?

fisico30

 

[tiny-tim]

oooh … page 3 !

Every point inside an extended object has 3 principal axes of rotation associated to it. For instance, point A has 3 orthogonal principal axes. Point B also have its 3 principal axes (with different eigenvalues than the eigenvalues of point B). The triad for point A can be oriented differently than the triad for point B, correct?

no, the orientation is always the same …

the 3 principal axes at different points are always parallel

Is the triad of principal axes passing through the center of mass the one that has the smallest eigenvalues among all sets of eigenvalues, hence the smallest moments of inertia?

yes … since each eigenvalue is the moment of inertia about that eigenvector as axis, and since moment of inertia satisfies the parallel axis theorem I = I_c.o.m + md²,

so it follows that the point with the smallest eigenvalues will be the centre of mass

For the same force, why, physically an conceptually speaking, does a longer lever arm increase the ability to cause a change in rotation? Why does a force that is applied farther from the pivot point cause a larger effect? Has this phenomenon been discovered empirically?

uh-uh, this is a distinct topic, and there's several threads on it already …

(my preference is to use the principle of work, but other people do it other ways)

read some of them, and if you still want to discuss it, start a new thread!

 

[fisico30]

hello,
I am trying to stay on the same topic. If the object that we throw in the air has a non symmetric distribution of its mass, the CM will not coincide with the geometric center of the object.
For instance, a rake, thrown in the air should be subject to a torque due to the force of gravity and spin faster as its flies...
Right?

As far as the why the moment of a force works the way it works, I can see how the work principle explains it. But that is still too mathematical for me. If we push with a force at a distance from the pivot, the larger the distance the more the ability to cause rotation. From a purely conceptual point of view, what does that distance do? Maybe it was an empirical discovery that does not have further explanation...
The definition of work helps mathematically but the situation is the same as trying to explain gyroscopic precession: the math is clear but why it works that way is less. It is possible to explain it however (I did find a good resource for that).


thanks
fisico30

 

[jbriggs444]

hello,
I am trying to stay on the same topic. If the object that we throw in the air has a non symmetric distribution of its mass, the CM will not coincide with the geometric center of the object.
For instance, a rake, thrown in the air should be subject to a torque due to the force of gravity and spin faster as its flies...
Right?

No. Ignoring tidal forces, gravity does not induce torque. The tidal forces due to one end of a garden rake being four feet farther from the center of the Earth than the other are negligible.

[Edit -- does not induce torque about the center of gravity]

You can balance the rake on your finger if you put your finger just under the center of gravity. That's a pretty good demonstration that gravity does not induce a torque. The human race has been using this fact for several thousand years when weighing goods for commercial purposes.

 

[fisico30]

but when the rake is flying in the air the center of mass has no support under it to balance it.
Sure, the weight force acts on the center of mass.

Do we assume that the object, if it rotates, must always rotate about the center of mass? In that case weight exerts zero torque...

thanks
fisico30

 

[tiny-tim]

Do we assume that the object, if it rotates, must always rotate about the center of mass? In that case weight exerts zero torque...

no, it doesn't rotate about the centre of mass …

it can only do that if the centre of mass is (instantaneously) stationary!​

yes, gravity exerts zero torque, and so it cannot affect the angular velocity … if you flip something into the air, whatever angular velocity it has when you let go, it keeps (while gravity makes the centre of mass move along a parabola)

 

[fisico30]

Thank you again.

So, about what point does the object flying in the air rotates about?
If we were on the object, on which point would we seat and see everything else rotate around us while we translate along the parabolic trajectory?

Tiny-tim, you mention that it can only do that if the centre of mass is (instantaneously) stationary!.

Do you have an example? When would the CM be instantaneously at rest from the lab reference frame?

thanks
fisico30

 

[tiny-tim]

So, about what point does the object flying in the air rotates about?

could be anywhere

eg if it was thrown vertically upward (while rotating), the instantaneously stationary point about which it instantaneously rotates would be somewhere on the horizontal line through the centre of mass

If we were on the object, on which point would we seat and see everything else rotate around us while we translate along the parabolic trajectory?

i'm not sure what you mean

if we were at the centre of mass, we would see the whole object rotate around us, while we follow a parabola

Tiny-tim, you mention that it can only do that if the centre of mass is (instantaneously) stationary!.

Do you have an example? When would the CM be instantaneously at rest from the lab reference frame?

in the lab frame, never, unless the motion is purely vertical, and the object is exactly at the top of its flight

 

[fisico30]

Ok, so you call "instantaneous" the point of rotation (I guess we should talk about the axis of rotation) because it changes with time with respect to the lab frame?

But from the point of view of the flying object, the only point that is not rotating is the CM...
that is why I continue to call that the center of rotation of the unconstrained flying object..

fisico30

 

[tiny-tim]

Ok, so you call "instantaneous" the point of rotation (I guess we should talk about the axis of rotation) because it changes with time with respect to the lab frame?

yes

But from the point of view of the flying object, the only point that is not rotating is the CM...
that is why I continue to call that the center of rotation of the unconstrained flying object..

i see what you mean, and that's certainly a valid way of describing the motion (the centre of mass follows a parabola, and the body rotates about it), but it isn't actually the centre of rotation

 

[fisico30]

Tiny-tim,

when I asked about the orientation of each triad of principal axes associated to each point inside an extended object, you replied that principal axes associated to different points are parallel to each other...How do you know that? Where can I find a proof of it? Goldstein book maybe?

My basic mechanics book don't mention that interesting aspect...

thanks
fisico30

 

[fisico30]

As far as the center of rotation and axis of rotation, if we are at the origin of our frame of reference (relative to which we are at rest), and a car drives by, the car is rotating about our location. In fact the car, even if it is moving in a straight line, has angular momentum
p=mvr sin(theta).

So, what is the axis of rotation? It is a straight line (can it be curved? Maybe so, in deformable objects) formed by points that are at rest relative to other moving points.
In a sense, every point or points that are attached to the frame of reference represent centers of rotation...do they?

thanks
fisico30