# Why do objects always rotate about their centre of mass?

• Leo Liu
In summary, the theorem states that the rotation of a rigid body will always be around its center of mass. This is true for any external force, regardless of its nature.
John Mcrain said:
If ask like this; in what point passes axis(perpendicular to water surface) of circular motion of the boat?
Is now answer only point A?
Who says that it is circular motion? It could be cycloidal or helical as well.

But you still have not defined what you mean by a center of rotation.

jbriggs444 said:
Who says that it is circular motion? It could be cycloidal or helical as well.

But you still have not defined what you mean by a center of rotation.
White path at sea is circular, at the center of that circle passes axis of boat "circular motion"

John Mcrain said:
White path at sea is circular, at the center of that circle passes axis of boat "circular motion"
The trajectory is only circular when projected onto a two dimensional surface in a particular way.

Now you have to figure out how to nail down the details of that projection.

jbriggs444 said:
The trajectory is only circular when projected onto a two dimensional surface in a particular way.

Now you have to figure out how to nail down the details of that projection.
In my task boat circle motion is known not something that I must find out..

John Mcrain said:
In my task boat circle motion is known not something that I must find out..
Nobody but you is discussing the motion. Everyone else is discussing how to describe the motion.

John Mcrain said:
If ask like this; in what point passes axis(perpendicular to water surface) of circular motion of the boat?
Is now answer only point A?
This is a good start to a definition for the center. The velocity defines a unique direction in a given frame. Perpendicular to the velocity defines a plane. How are you planning on picking out a specific point in that plane? We don’t want to use the water since there are many scenarios with rotation but without water.

You see, it is not a trivial exercise to define the center.

My answer is I dont know..and dont understand what is the point

John Mcrain said:
My answer is I dont know..and dont understand what is the point
The point is that we choose the center of rotation (rather arbitrarily) in order to simplify the translation. The center of rotation is not something that is part of the world, it is part of the analysis.

Dale said:
The point is that we choose the center of rotation (rather arbitrarily) in order to simplify the translation. The center of rotation is not something that is part of the world, it is part of the analysis.
But if we relativize center of rotation/circular motion, that mean in real life this point dont exist in space.
I mean boat really circle around point A, if you swimm at point A you will see that boat moves around you.
I dont understand this part.

Dale said:
If they did not then the center of mass would not be traveling in a straight line. This would violate Newton’s first law.

Dale said:
The point is that we choose the center of rotation (rather arbitrarily) in order to simplify the translation. The center of rotation is not something that is part of the world, it is part of the analysis.

Last edited:
Dale said:
The point is that we choose the center of rotation (rather arbitrarily) in order to simplify the translation. The center of rotation is not something that is part of the world, it is part of the analysis.
Indeed, you can choose any body-fixed reference point together with any body-fixed (Cartesian) basis to describe its motion. Of course you can choose a very stupid reference point to make simple problems unsolvable (or at least very hard to solve).

Take a very simple example of the motion of a rigid body around a fixed axis. Of course, here you should choose the body-fixed origin of the body-fixed reference frame on this axis. In principle you could choose any other body-fixed origin, but then you get very complicated equations of motion. The proper choice of your reference frames is a big step towards the solution!

Dale
John Mcrain said:
But if we relativize center of rotation/circular motion,
These are two different things. You are confusing yourself by lumping them together. Rotation (changing orientation) and motion along a circle (translation) are two independent components of motion. You can have one without the other.

John Mcrain said:
if you swimm at point A you will see that boat moves around you.
Yes, A is the static point in the frame of the water that the boat translates around along a circle. But the rotation of the boat (changing orientation) is a separate issue, and the point A is not the only point that can be used as reference for rotation.

As I tried to explain in #115 of course the choice of the body-fixed (which is usually non-inertial/rotating of course) as well as the space-fixed (usually inertial) reference frames is completely arbitrary, while of course the motion is uniquely determined by the dynamics and the initial conditions. Usually there's a "natural choice" for these reference frames, i.e., given by the physical situation you choose a convenient body-fixed origin and (Cartesian) basis for your body-fixed reference frame (e.g., given the origin the basis should chosen along the principle axes of the tensor of inertia, to make its body-fixed components diagonal).

A nice example is a cylinder with an off-axis center of mass. This you can describe using either the body-fixed origin on the axis or in the center of mass. You get of course the same results for the motion as a whole, but the decomposition into "translatorial" and "rotatorial" motions is different. See Sect. 4.3.3 in

https://itp.uni-frankfurt.de/~hees/publ/theo1-l3.pdf

(in German; if I find the time, I'll put it to my English translation of the rigid-body chapter of this manuscript, part of which can be found here:
https://itp.uni-frankfurt.de/~hees/pf-faq/spinning-top.pdf
but presently contains only the basics of the theory of the spinning top).

A.T. said:
These are two different things. You are confusing yourself by lumping them together. Rotation (changing orientation) and motion along a circle (translation) are two independent components of motion. You can have one without the other.Yes, A is the static point in the frame of the water that the boat translates around along a circle. But the rotation of the boat (changing orientation) is a separate issue, and the point A is not the only point that can be used as reference for rotation.
Boat moves in circle around point A, but boat do not rotate(change orientation) because allways same side of boat looking toward point A.

Boat dont spins around itself.

John Mcrain said:
Perhaps I shouldn’t have said “violates” Newton’s first law in the first. Something like “complicates” would have been better.

The point I was trying to make in that first quote is that the key to pay attention to is how that choice of center affects the translation. We want a center that simplifies the translation. That is the same point I am making later.

vanhees71
John Mcrain said:
Boat moves in circle around point A, but boat do not rotate(change orientation) because allways same side of boat looking toward point A.

Boat dont spins around itself.
Well that took a turn I didn't expect! This coordinate isn't a normal rectangular/Cartesian coordinate system centered on point A then, since the boat points up, then left, then down. You're trying to use polar coordinates.

...but it turns out that's not sufficient. Rotation is absolute. You can measure it independent of other motion. The boat is rotating at a rate of one rotation per revolution (like the moon).

Dale and vanhees71
John Mcrain said:
Boat moves in circle around point A, but boat do not rotate(change orientation) because allways same side of boat looking toward point A.

Boat dont spins around itself.
This is simply incorrect.
If you don't see it then you don't see it.
For the boat not to rotate, the prow would need to point at a fixed point very far away....say Bangor Maine. Does your boat always point at Bangor?

Dale and vanhees71
hutchphd said:
This is simply incorrect.
If you don't see it then you don't see it.
For the boat not to rotate, the prow would need to point at a fixed point very far away....say Bangor Maine. Does your boat always point at Bangor?
But it's easy to see. Just put your body-fixed coordinate system (body-fixed origin and Cartesian body-fixed basis) and draw it for different times. You'll see that the body-fixed basis rotates relative to the space-fixed (inertial) basis. Only the decomposition of the motion in translatiional parts (i.e., the motion of the body-fixed origin relative to the space-fixed origin) and rotational parts (i.e., the motion of the other points of the body relative to the body-fixed origin) changes by changing the body-fixed origin. The overall motion is of course the same irrespective of the choice of any reference frames.

Dale
John Mcrain said:
Boat moves in circle around point A, but boat do not rotate(change orientation)
In the rest frame of the water, which you claimed to use, the boat does change orientation.

John Mcrain said:
because allways same side of boat looking toward point A.
In the rest frame of the water, the vector (boat -> A) changes orientation as well.

russ_watters and vanhees71
A.T. said:
In the rest frame of the water, which you claimed to use, the boat does change orientation.In the rest frame of the water, the vector (boat -> A) changes orientation as well.
Yes it will not change orientation(rotate) if bow point allways to the north.

John Mcrain said:
Yes it will not change orientation(rotate) if bow point allways to the north.
Right, but is not likely to be case here, so the boat is rotating. Around which point is a matter of choice.

Lnewqban
A.T. said:
Right, but is not likely to be case here, so the boat is rotating. Around which point is a matter of choice.
I agree. one rotation per one full circle.

Is point about boat travel in circular path also metter of choise or it only can be point A?

Sticking with boats: Suppose two identical boats are tied with a rope at the gunnels (pointing opposite directions). Each boat will travel in a circle about the centerpoint of the rope (point A). But each boat captain will correctly rerport also that the other boat is circling about his boat. And that his own boat is rotating about its center!
They are all correct descriptions.....So not really specific enough. Can we be finished now?

Dale
John Mcrain said:
Is point about boat travel in circular path also metter of choise or it only can be point A?
If you want a center of the circular translation that is static in the rest frame of the water, then it can only be A.

John Mcrain
A.T. said:
If you want a center of the circular translation that is static in the rest frame of the water, then it can only be A.
Finally came to mine

John Mcrain said:
Finally came to mine
Notice he said center of circular translation, not center of rotation.

A.T., russ_watters and John Mcrain
Dale said:
Notice he said center of circular translation, not center of rotation.
It will be easier to learn with examples of solving tasks or animations, not with words.
For example after I watched this animation I figure out why my boat and moon rotate as well.

John Mcrain said:
It will be easier to learn with examples of solving tasks or animations, not with words.
It will be easier for you. The statement is otherwise presumptuous.
For many technical trained people, the math is by far the easiest. Vectors make the entire world far more comprehensible.

John Mcrain said:
Finally came to mine
You may not be aware, but you've created a potential contradiction for yourself here by having two different axes. One for the rotation and a different one for the translation.

John Mcrain said:
It will be easier to learn with examples of solving tasks or animations, not with words.
For example after I watched this animation I figure out why my boat and moon rotate as well.

Note that by limiting yourself to a special case where rotation and circular translation have the same frequency you are setting yourself up to confuse them. You should look at other cases to understand the difference between them and their possible centers.

russ_watters

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