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Being a high school student, I read the concept of center of mass and it was written in my book that

When a spinning ball is projected with some velocity , then all the points on the ball have complicated paths except the center of that ball which follows the well known parabolic trajectory. And hence we define that point as center of mass.

However, I think that all the points on the any axis about which the ball is spinning follow the parabolic trajectory and are not influenced by spin .

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Edit : Most of the answers argued that the rotation axis may change because of torque but the main point to note here is that we can't differentiate between two axis in case of a sphere since it is symmetric from all sides and also that a sphere can't rotate about more than one axis at a time . So saying that it will rotate about different axis is I think meaningless.

So is it okay to define axis of mass in place of center of mass for sphere or other symmetric bodies or am I wrong somewhere ? If not, give a proper reason.

Ankit
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    A ball can spin around multiple axes simultaneously. – OrangeDog Oct 25 '20 at 14:48
  • @OrangeDog how can this be true ? See this :https://physics.stackexchange.com/a/322274/271783 – Ankit Oct 25 '20 at 14:52
  • That says central axes. The Moon, for example, is spinning both around its polar axis, and around the Earth. The point being, all points on your "axis of mass" other than the center can still be moving not on a parabolic trajectory. – OrangeDog Oct 25 '20 at 15:35
  • Isn't everyone over-engineering a much simpler point? If it's a properly balanced sphere, how could any movement change its centre of mass, or force - or allow - that centre to extend into an axis? – Robbie Goodwin Oct 25 '20 at 21:22
  • I think you should probably change your question for a more specific title – tryst with freedom Oct 26 '20 at 18:25

10 Answers10

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Your intuition is correct. All the points along the axis of rotation share the same motion. In fact, defining the center of mass by means of the rotation axis isn't sufficient.

It is possible to define the center of mass as the only point where all possible rotation axis pass through for a freely rotating rigid body. So you have to consider all possible lines passing through the center of mass (called a pencil of lines) to define the center of mass.

This is a consequence of Newton's second law and the concept of momentum. The definition of momentum for a collection of particles leads to the definition of center of mass as being the special point whose motion and combined mass can be used to describe all the individual contributions of momentum from each particle.

$$ \vec{p} = \sum_i m_i \vec{v}_i = \left( \sum_i m_i \right) \vec{v}_{\rm COM} = m \, \vec{v}_{\rm COM} $$

There is only one point on a rigid body that satisfies the above. And the condition for this point is the definition for center of mass

$$ \sum_i m_i \vec{r}_i = \left( \sum_i m_i \right) \vec{r}_{\rm COM} $$ or $$ \vec{r}_{\rm COM} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i} $$

PS. In geometry, a point is equivalent to a pencil of lines through that point. Just like a line is equivalent to a locus of points along the line.

PS2. See also this similar answer to a related question.

PS3. Here is an overview of the development of equations of motion for a rigid body. This might be advanced reading at this point, but it contains all the concepts needed to understand the subject.

John Alexiou
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  • why isn't it sufficient to define center of mass by means of rotation axis ? I have added some points to the question. Hope you respond accordingly – Ankit Oct 26 '20 at 12:03
  • @Ankit Because an axis defines a locus of points and not a unique point. The center of mass is a unique point and its definition must reflect that, – John Alexiou Oct 26 '20 at 12:13
  • why do we need to define a point and not an axis ? – Ankit Oct 26 '20 at 12:45
  • Because the center of mass is a point, even for a sphere. It is a point because it represents all axes that could pass through it as free rotation axes. – John Alexiou Oct 26 '20 at 16:19
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The idea of a center of mass even works for a "disconnected" system of particles. You can take any set of particles, call it your system and talk about its center of mass. The equation of the center of mass would still be valid, even if all the particles are moving chaotically and there's no axis to talk about for the entire system.

Even for rigid bodies, the "axis of mass" wouldn't be a fixed line. The axis of rotation could keep changing depending on how the external torques add up vectorially. So the "axis of rotation" wouldn't be a property of the rigid body, but rather something which arises in a particular situation. The axis of mass could literally be any line passing through the center of mass.

EDIT- Responding to your edit, I think the whole point of your post was that, since any point on the axis of rotation can be seen as following a simple path throughout the motion of the ball, so all the axis points are as special as the CoM.

However, this is not true. The other points on the axis are NOT following a simple path in general. In the most general case of a direction changing torque (and hence a continuously changing axis of rotation), any other point (other than the CoM) on any of the instantaneous axes of rotation, is not following a simple path throughout the motion of the sphere. More importantly, the equation $a_{point}=\frac{1}{M}\vec{F_{ext}}$ is only valid when the point is the CoM. This equation is what's responsible for the CoM to follow a simple path. The motion of any other point is affected by the internal as well as the external forces in general.

As for "why don't we define a general diameter of the sphere as the set of all possible axes of rotations?", that'd be useless as any line passing through the CoM of a rigid body can behave as an axis of rotation. The axis of rotation at an instant is entirely dependent on the direction of the instantaneous angular momentum, which is again dependent on the external torques. The CoM, on the other hand, is a fixed property of the rigid body. This is why you study things like: CoM of a circular disc, CoM of a uniform cylinder, etc, instead of the axis of rotation of a circular disc.

I think you might see some sentences in your physics book like "The axis of rotation is always some line passing through the center of mass (unless there's a rotation about a forced axis)". That is all there is to the idea of "axis of mass".

Ryder Rude
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  • depending upon the system of particles which follow simple path why is it improper even if this axis precess ? – Ankit Oct 24 '20 at 10:43
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    @Ankit - it's not that's "improper", it's that a specific axis only applies to a particular case, and not always. So it can be useful in a particular situation, but for general laws of physics (when discussing things that are simultaneously true for all kinds of different situations), the idea of the center of mass (COM) is more useful. Besides, COM is an intrinsic property of a body. E.g. you can take any rigid body and find its COM (a single exact point) even if it's standing still (there's a way to do it which involves calculus). – Filip Milovanović Oct 24 '20 at 21:38
  • @Ankit You said that the entire axis of mass follows a simple path for a rigid body. However, that is not true in the general case where the direction of torque keeps changing. The points which lie on the "axis of rotation" at one moment need not lie on the "axis of rotation" at the next moment. CoM is the point which is common to all those axes of rotations . So in general, only the CoM will be the point following a simple path even for rigid bodies – Ryder Rude Oct 25 '20 at 02:39
  • @Ryder Rude are you talking about precession ? – Ankit Oct 25 '20 at 02:46
  • @Ankit Yes. Keep in mind that the "axis of mass" is not a property of the body. The axis is completely dependent on the external torques applied. – Ryder Rude Oct 25 '20 at 02:49
  • @Ankit In special cases where the axis is fixed throughout the motion, physics books call it the "axis of rotation" instead of "axis of mass". – Ryder Rude Oct 25 '20 at 02:51
  • @Ryder Rude I don't think that points on the axis keep on changing due to precession . Although the axis itself rotates but it doesn't slice the sphere i.e the sphere also wobbles with it and thus points remain the same . – Ankit Oct 25 '20 at 02:51
  • @Ankit I think I misunderstood the term "precession". Precession seems to be a very special case of what I'm saying. I'm talking about when you apply a torque on a rotating body, the torque having a different direction than the rotation axis of the body. It'll cause a change of axis. The points on the previous axis will suddenly start following a complicated path. The CoM will be the only point following a simple path throughout the motion. – Ryder Rude Oct 25 '20 at 02:57
  • @Ankit Try rotating a ball about one axis first (some diameter of the ball). Then try to introduce another rotation to it about some other axis (some other diameter of the same ball). The physics says that the ball will start rotating about a third diameter instead, which will be like the vector sum of the initial diameter and the new diameter you tried to introduce.Throughout this experiment, the points on the first diameter will initially be at rest, but will later start going in circles (when you changed the axis to another diameter).The CoM is common to all diameters and will stay at rest. – Ryder Rude Oct 25 '20 at 03:11
  • @Ruder Rude I have added some points to the question. Hope you respond accordingly. – Ankit Oct 26 '20 at 12:04
  • @Ankit I responded – Ryder Rude Oct 27 '20 at 12:21
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You are right in the case of a sphere. A tennis ball with top spin will not change to side spin during its trajectory for example.

But for less symmetric bodies things are different. If the player throws its racket instead of the ball, the rotation axis will (in general) change indeed. Only the CM follows the parabolic path.

Claudio Saspinski
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  • so can we define axis of mass for a sphere ? – Ankit Oct 24 '20 at 15:38
  • If your definition of centre of mass was that points that don't move due to internal motions, then axis of mass would exist (for rotation. it need not be sphere but anything) – Rishab Navaneet Oct 24 '20 at 16:20
  • But our definition for centre of mass is the point where all the masses seem to be concentrated. That is if we apply a constant force all over the body, then where would it effectively act. Or in another way, which is that point which does not move until there is an external force on it. Again that means the point which does not move even if the entire mass is wobbling – Rishab Navaneet Oct 24 '20 at 16:23
  • "If the player throws its racket instead of the ball, the rotation axis will (in general) change indeed." Not quite.. If the rotation axis is the first or third principal axis, it doesn't change axis. If the rotation is exactly the second principal axis, it doesn't change. If the rotation is around the second principal axis plus a perturbation, that perturbation gets magnified. – Acccumulation Oct 24 '20 at 20:14
  • @Acccumulation Agree That is why I wrote (in general). – Claudio Saspinski Oct 24 '20 at 21:14
  • @Ankit - when solving problems, you often want to ignore rotation (e.g., you just care about the object's path), so the concept of center of mass is more generally useful (you can apply it to a lot of different scenarios). E.g., even for a sphere, when it hits something, for example, the rotation axis will change, but it will still have the same center of mass - and you can use it to describe its motion. Now, in more specific applications, e.g., in a video game physics simulations, you may have something like (center of mass & current rotation axis) stored as data. – Filip Milovanović Oct 24 '20 at 21:25
  • @Filip Milovanović I think the position of center of mass also changes since nothing is perfectly rigid and everything get compressed. – Ankit Oct 25 '20 at 02:18
  • @Ankit - Yes that's true; in physics we often talk about idealized things, but if you can deform the body, you can change its distribution of mass. But here's what's cool about the COM; if you, say, hit a ball and send it flying, it will compress first, and then lift of and wobble in flight, but this wobbling will not affect the trajectory & momentum of the COM (only external forces can). Or consider a missile in flight, and then exploding mid-air - the explosion will send the debris everywhere, but in such a way so that the COM of the whole system continues on a parabolic trajectory. – Filip Milovanović Oct 25 '20 at 15:30
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Newton's laws as they are stated are only applicable for point mass bodies. To apply them for rigid bodies, we need a point on the body that we can consider all of its mass to be distributed in.

As per axis crossing center of mass, it's useful for a moment of inertia calculations (see parallel axis theorem) but it's not useful for talking about motion. It's not something which we can define consistently. One of the reasons is that out of all the axis passing through, you'd need to specify a new axis such that your equations hold.

For example, imagine the ball having a torque applied in a direction such that it is rotated around the $y$ axis, suddenly the axis you've shown in the question(assuming it is the z) won't work so well.

See this answer for more details

And, it may be good to know that this extension was actually done by Euler (read here)

Note: point of this answer was to highlight the importance of the center of mass.

tryst with freedom
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You got an axis full of points moving nicely because you considered rotation. It always happen along an axis and particles along that line does not move.

But consider a more complex system. Say a drop of water or hiighly elastic rubber ball which can deform, and compress when thrown, then there would only be one point which moves properly with a constant velocity.

Rishab Navaneet
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  • first of all I didn't understand your water drop example and can you please elaborate your second example or add a picture of that ? – Ankit Oct 24 '20 at 17:17
  • I think what @Rishab explained was much simpler systems not at all complex. – David Oct 25 '20 at 03:42
  • https://youtu.be/EzahpSqGbVg The drop here is an example of the wobbly system mentioned in answer. The individual molecules in the drop may move in complex ways but the centre of mass is that point which would not move at all Unless an external force acted. – Rishab Navaneet Oct 25 '20 at 10:13
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Your 'axis of mass' depends on how the ball is spun, which depends on the whim of the person throwing it. So it is not a property of the ball on its own. But the centre of mass follows a parabola however the spin is aligned.

RogerJBarlow
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If the ball has a single axis of rotation, then all points on that axis will not rotate (that's pretty much the definition of "axis of rotation"). However, it's possible for a ball to have more than one axis of rotation simultaneously, so while their claim is misleading in that it implies that all non-center points always have complicated paths, it is true in the sense that only the center of mass is guaranteed to have a simple path.

As for your question of whether we can define an "axis of mass", that is clearly impossible in the case of a sphere; since the spherical is ... well, spherically symmetric, there's no way to distinguish a particular axis. There is one axis distinguished by the rotation, but that axis is particular to that motion, and not an inherent property of the sphere. Even if there is a particular axis that it is rotating around, there are an infinite number of axes that it could be rotating around.

For objects in general, it is possible to distinguish three principal axes.

Acccumulation
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Take a globe. Spin it around its normal axis, the polar axis the way the Earth spins. Then pick the globe up and while it is still spinning the first way, flip it north to south the way you would a coin. There you have a sphere spinning over two axes at the same time and your axis of mass makes no sense. The center of mass and only the center of mass is traveling in the parabolic trajectory under your conditions.

Bill Watts
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If the sphere is fully symmetric, then you can't define "the" axis of mass, because ANY axis of the sphere would be equally good, so it's not a well-defined quantity. Only the center is well defined.

In general, in 3D Newtonian physics, an object actually has three "axis of mass," expressed by the Inertia Tensor. This is used to good effect in-game physics simulations, where a long, thin object tumbles differently from a short, fat object. This is also why objects typically end up spinning around one "main" axis of rotation (the "biggest" axis in the tensor) but can use off-center angular momentum to "flip" around one or both other axes. Google the "spinning corkscrew in zero gravity" YouTube videos for some illustrations.

Young Kindaichi
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Jon Watte
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In some circumstances, like you describe, you can construct an axis of mass. However, this is typically not done. One of the major reasons to operate on a center of mass is that it completely decouples the translational movement from the rotational movement. They can be handled independently, with independent sets of equations to solve. To have an axis of mass brings the rotational component back into the translational part of the problem, wrecking that independence.

In some easy cases you can do this. It just means you have to solve a more difficult problem instead of an easier problem. However, in more complex situations this gets very difficult very quickly. For example, if you have a gyroscope, it precesses and nutates, which means the axis of rotation is constantly changing. This means your axis of mass would have to change. However, if you did the calculations using a center of mass instead, this rotational effect would not matter when you're computing the translational results.

So in the end, in some situations you can think in these terms, but it often complicates the problem unnecessarily.

That being said, there is a related concept known as Screw Theory where we model the translation and rotation as one "screw like" motion and has a meaningful screw axis which is used for both translation and rotation. We typically do not teach it in physics class because it's mathematically quite a complicated way of thinking about it. However, it is popular in robotics because it turns all translations and rotations of an arm into screw multiplications. For the sorts of things they do in robotics, like inverse kinematics, the simplicity of treating everything as a multiplication outweighs the complexity of mixing together translation and rotation. However, the screw axis is not the same axis as the "axis of mass" you describe. It's a different axis.

Cort Ammon
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