Rotation around center of mass

[BrainSalad]

I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation. I also understand that the center of mass of an object of uniform density is located at the centroid. In this case, I understand that the center of mass follows a path identical to that of a point mass subjected to the same initial forces, and that any rotation is around the centroid/COM. This makes sense to me because of the conservation of linear momentum (the net momentum due to rotation is zero, so the object behaves as a point mass). However, an object with non-uniform density necessarily has a center of mass outside it's geometric center. In this case, if the object is subjected to a force and a torque, if the object is to maintain rotation at constant angular velocity around it's COM, momentum due to rotation cannot equal zero (the less dense parts of the object are located farther from the center of mass and thus have greater linear velocity). How can an object of non-uniform density maintain rotation at constant angular velocity around its center of mass if the direction of momentum due to rotation is constantly changing? It seems that the object would be pulled in different directions due to "uneven" linear momentum. Please correct my flawed reasoning if possible.

 

[mfb]

I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation.

Only if the object has a spherical symmetry (and only in 3 dimensions).

However, an object with non-uniform density necessarily has a center of mass outside it's geometric center.

No it does not. What about a spherical shell?

In this case, if the object is subjected to a force and a torque, if the object is to maintain rotation at constant angular velocity around it's COM

Why should it do that?

(the less dense parts of the object are located farther from the center of mass

They don't have to be (but they usually are in natural objects).

and thus have greater linear velocity).

Or smaller, depending on the side you consider.

How can an object of non-uniform density maintain rotation at constant angular velocity around its center of mass if the direction of momentum due to rotation is constantly changing?

Why do you think it does that?

Most natural objects have a mass distribution very close to a spherical symmetry. And the deviations actually lead to torques on them. The rotation of Earth is constantly slowing down due to this, for example. You don't see this on a scale of a human life (although it is possible to measure it), but within millions and billions of years the effects are significant. Some dinosaurs just had ~23 hours in a day.

 

[BrainSalad]

Are you saying that objects don't always rotate around their center of mass?

 

[mfb]

No I do not say that.

Objects in free space can always be described with a rotation around the center of mass together with a uniform motion of the whole object.

 

[D H]

I understand that the center of mass is a point which can be considered to contain all of an object's mass, for the purpose of calculations involving universal gravitation.

That is incorrect. It is true if the object has a spherical mass distribution. It is approximately true if the point in question at which one wishes to calculate gravitational attraction toward some object is far removed from the object.

I also understand that the center of mass of an object of uniform density is located at the centroid. In this case, I understand that the center of mass follows a path identical to that of a point mass subjected to the same initial forces, and that any rotation is around the centroid/COM. This makes sense to me because of the conservation of linear momentum (the net momentum due to rotation is zero, so the object behaves as a point mass).

The translational behavior of *any* rigid body subject to some net force is identical to that of a point mass of the same mass as the rigid body that is located at the center of mass of the rigid body and that is subject to the same net force as the rigid body.

However, an object with non-uniform density necessarily has a center of mass outside it's geometric center.

This also is incorrect.

In this case, if the object is subjected to a force and a torque, if the object is to maintain rotation at constant angular velocity around it's COM, momentum due to rotation cannot equal zero (the less dense parts of the object are located farther from the center of mass and thus have greater linear velocity). How can an object of non-uniform density maintain rotation at constant angular velocity around its center of mass if the direction of momentum due to rotation is constantly changing? It seems that the object would be pulled in different directions due to "uneven" linear momentum. Please correct my flawed reasoning if possible.

You are assuming angular velocity is a conserved quantity. It isn't. It's angular momentum rather than angular velocity that remains constant if no external torques act on the object.

Are you saying that objects don't always rotate around their center of mass?

You can pick any arbitrary point on an object as a fixed point such that the object's behavior is modeled as combination of a translation of that fixed point and a rotation about it. The key reason for choosing the center of mass as the fixed point is because this is the unique point where the translational and rotational equations of motion decouple. Nonetheless, there are times when it makes more sense to choose some other point as the fixed point.

 

[BrainSalad]

What does it mean for rotational and translational motion equations to decouple?

 

[mfb]

If they decouple, they get independent of each other.