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I read this question on why objects tend to rotate around their center of mass.

The accepted answer said that objects in general do not rotate around their center of mass.

I'm confused about this. I've seen derivations that conclude that the sum of forces acting on a system of masses (like a rigid body) divided by the mass of the system equals the acceleration of the center of mass of the object.

Now let's apply a set of forces on an object. The object is free, that is, it is not bound by any kind of hinge. We have concluded that the center of mass of the object will start accelerating. Since the forces are constant, the acceleration of the center of mass is constant. But if the acceleration is constant, the acceleration should be linear and the COM should be moving in a straight line, and therefore not be rotating. But the accepted answer on the question disagrees. So what am I getting wrong?

And if this unbound object does not rotate around its COM, around what point does it rotate? If the have an object floating in space (like a straight rod), and we apply a force to one of its ends, how do we determine the point around which it starts rotating?

I also found this website.

It states that a thrown knife rotates around its center of mass. This seems to agree with my reasoning: When flying in the air, all parts of the knife experience gravitational force. These forces summed divided by the mass of the knife gives the acceleration of the COM of the knife. From projectile motion we know thrown object fly in parabolic paths. Therefore, the COM of the knife flies in a parabolic path, and so it cannot be in rotation.

So could somebody shed some light on this matter? When do objects tend to rotate around their centers of mass?

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  • $\begingroup$ Simple rotations are about an axis, and when the object is isolated and rotating its rotation axis will pass through the center of mass, as with earth for example, A rotation induced by external forces can have an axis anywhere through the body. $\endgroup$ – anna v Mar 29 at 11:28
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    $\begingroup$ Have a look at this answer and Appendix 20A Chasles’ theorem $\endgroup$ – Farcher Mar 29 at 12:23
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There are two aspects to this question. One is about geometry, and the other is about mechanics.

From a geometrical point of view, at every instant in time, any motion of a rigid body in 2D space that involves rotation is equivalent to a rotation about some fixed point in space. For example, at any instant a car wheel is rotating about the fixed point in contact with the road. Of course the "fixed" point is different at each instant in time, as the car moves.

On the other hand, when doing mechanics it may not be very interesting to know which point is "fixed," especially if the "fixed" point is not actually inside the object. Knowing the position of the "fixed" point is only important if there are some constraints on how the system can move, which apply some forces at the fixed point to make it a fixed point - in the car wheel example, the normal (weight) and tangential (friction) forces between the wheel and the road.

If there are no such forces, the simplest way to describe the motion in Newtonian mechanics is as a rotation about the center of mass, plus the translation of the center of mass. That is because the equations for the translation of the COM and the rotation about it are independent of each other, so they can be solved separately.

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Freely moving objects rotate around their center of mass, think about anything from a ball to a stick, and the you throw it. It definitely rotates around its center of mass and there is a proof for that that I'm not going to develop here.

But then, if there is an external force applied ir can rotate around any axis. Think about a stick rotating along an axis that is set in one end.

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  • $\begingroup$ How do we define freely moving objects? You say that thrown objects rotate around their center of mass, but also that under external forces objects can rotate around any axis. But when thrown, arent objects under an external force, gravity? $\endgroup$ – S. Rotos Mar 29 at 12:33
  • $\begingroup$ Just an object moving through space without any force acting on it. $\endgroup$ – Ballanzor Mar 29 at 12:34
  • $\begingroup$ So a thrown object on Earth does not rotate around its COM? $\endgroup$ – S. Rotos Mar 29 at 12:41
  • $\begingroup$ Well, yes it does. You need to take into account torque. Since you can think of the gravitational force as a force applied in the center of mass, there is no torque. $\endgroup$ – Ballanzor Mar 29 at 12:45
  • $\begingroup$ Without the development, this is a comment, not an answer. $\endgroup$ – garyp Mar 29 at 14:01
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Now let's apply a set of forces on an object. The object is free, that is, it is not bound by any kind of hinge. We have concluded that the center of mass of the object will start accelerating. Since the forces are constant, the acceleration of the center of mass is constant. But if the acceleration is constant, the acceleration should be linear and the COM should be moving in a straight line, and therefore not be rotating. But the accepted answer on the question disagrees. So what am I getting wrong?

I think this answers your question fine. In order for there to be rotation about the center of mass the center of mass cannot be changing direction. Therefore, you need constant forces acting on your object. If the forces acting on the object just happen to exert torques about the center of mass then we will have rotations as well.

Contrast this with something like a rod on a hing falling from some initial height. The force supplied by the hinge is constantly changing while the rod falls. This causes the center of mass to not move in a single direction and the rod will move in such a way that the end of the rod at the end of the hinge remains stationary.

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