Why do objects always rotate about their centre of mass?

[russ_watters]

Around what boat rotate?

As many others have said, you can choose anything you want. Me personally, since a speedboat's direction of motion isn't well tied to the direction it is pointing, I'd prefer describing the rotation as being about the center of mass (or helm). Especially if I'm the one driving it.

 

[A.T.]

Camera show reference frame of water, so I answer for this frame.

It's a still image, so you cannot tell anything about the reference frame of the camera.

You mean roatition is spin around point(cm) inside object ,and translation along circular path is around point outside of object?

That is one possible way to describe the motion.

Isnt this game of words?

To communicate you need to agree on what words mean. Centripetal force on an object doesn't imply that the object is rotating (changing orientation), just that it's center of mass is translating along a non-linear path, in inertial frames of reference.

 

[Dale]

Isnt this game of words?

Yes! That is the issue indeed. You are making a claim that there is some unique point (but you cannot define it). We are telling you that it depends on how you define things.

That is indeed the point of my challenge to define what you mean with full mathematical rigor. If you do so you will find that there is an element of choice in the definition or that it doesn’t produce a unique point.

 

[snorkack]

Bodies do not rotate about a point. They rotate about an axis.

Why does the axis contain the COM? It doesn't have to, but that's usually what we mwan by the word "rotation". The earth rotates about its axis (which contains the COM) but revolves around the sun. It's a convention to aid discussion.

Now one might say "yeah, but this is just terminology" and it is. But when trying to understand something, it's better to keep the potential miscommunications to a minimum.

Let´s consider an example where discussing "axis" which does NOT contain the centre of mass is the natural convention...
Out of balance wheel. What is an out of balance wheel rotating around - the axis which is being held fixed, or some imaginary "axis" which goes through its centre of mass and moves around with it?
Now, a wheel still looks close to symmetrical. How about something conspicuously asymmetric to reflection?
Like a steelyard balance beam? One end thick and heavy. The other end thin and long.
How do you balance a steelyard balance beam?
In an uniform field of gravity, first find an axis around which the moments of two arms are equal. This way, the beam will not press against the spot whereby it is held horizontal, and stays horizontal when released to be supported by axis only.
But this has only given you the location of centre of mass along the beam. If your axis is above CoM, the beam returns to horizontal when slightly displaced. If your axis is below CoM, the beam tips over when slightly displaced from horizontal.
Thus adjust the axis position vertically until the beam rotates freely and will stay balanced in any angle from horizontal. This shows that the axis goes through CoM.
Now how about rotation?
When the beam is steady in uniform field of gravity, the momentum is integral of mass times leverage (times gravitational acceleration). But when the beam is rotating, the centrifugal force is integral of mass times leverage (this time times square of angular speed).
Therefore, if either of them balances, both do - and vice versa. Fix a beam, or any body, on an axis that does not pass through centre of mass, and when it rotates, it will exert centrifugal force on the axis that varies with the direction of the body. And at some position, the gravity will have momentum relative to axis. Pick the axis through CoM, and the centrifugal forces will balance, and the weight of the body will be constant and applied direct on the axis.

 

[John Mcrain]

Yes! That is the issue indeed. You are making a claim that there is some unique point (but you cannot define it). We are telling you that it depends on how you define things.

That is indeed the point of my challenge to define what you mean with full mathematical rigor. If you do so you will find that there is an element of choice in the definition or that it doesn’t produce a unique point.

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?

 

[jbriggs444]

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.

 

[John Mcrain]

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"

 

[jbriggs444]

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.

 

[John Mcrain]

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..

 

[jbriggs444]

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.

 

[Dale]

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.

 

[John Mcrain]

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

 

[Dale]

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.

 

[John Mcrain]

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.

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

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.

Isnt this two answers in contradiction?

 

[vanhees71]

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!

 

[A.T.]

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.

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.

 

[vanhees71]

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).

 

[John Mcrain]

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.

 

[Dale]

Isnt this two answers in contradiction?

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.

 

[russ_watters]

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).

 

[hutchphd]

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?

 

[vanhees71]

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.

 

[A.T.]

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.

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.

 

[John Mcrain]

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.

 

[A.T.]

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.

 

[John Mcrain]

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?

 

[hutchphd]

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?

 

[A.T.]

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]

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

 

[Dale]

Finally came to mine

Notice he said center of circular translation, not center of rotation.

 

[John Mcrain]

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.

 

[hutchphd]

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.

 

[russ_watters]

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.

 

[A.T.]

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.