Expert: Daniel Mazur - 7/1/2009

Question
QUESTION: Hi Dan,

Another question I was wondering about and figured out an explanation to myself. Would like you to tell me if I'm wrong, and how, in my explanation.

Let us consider another thought experiment. I'm an astronaut and on the International Space Station. I wear a space suit, attach myself to the station and go out for a space walk and an experiment.
I also hav with me a plank of wood. It's a square plank, that has a uniform mass density.

Now I keep the plank in front of me so that any of it's two surfaces face me. Now it touch it with my finger at the center of mass, that is the very center point of this square in this case.
What I will observe is that this whole plate of wood will start accelerating away from me for as long as I touch it and will move with uniform velocity away from after I withdraw my finger.

So it obeys newton's second law.

So far, it's all good.

Now suppose i do the same thing again, with the only difference that this time I touch the plank not on the COM, but at a point a distance "X" away horizontally from the COM, on my right hand side.
Now the plank will also move forward the way it did in the first case, but now it will also rotate from my right to left.

My question is, why does it have to "rotate" in the second case. In the second case too, I applied a force and it should just accelerate. It does that but it also rotates. why??

The explanation I figured out is this:

I apply the force, or my finger is in touch with only a point on the entire plank, so it's this plank that experiences the force, and it's only this point that should accelerate. But this point is also electromagnetically bonded to the mass in it's surroundings. These bonds are strong and suppose the force I apply caannot break them and thus this point does not fly apart from the plank.
So i apply a force on this point. This point in turn applies the same force on the mass present to it's left and right. Now since the mass on the left is more than that on the right, the mass on the left accelerates less than the mass on the right of this point. So the entire thing rotates from my right to left.

Also i figured out intutively that the axis of rotation should be X units on the other side of the COM since the point of application of force is X units from COM on the right hand side.
i.e. the point and the axis are symmetrical about the COM.
I don't know if this is right, but that's what my intution tells me.
If it is right, how does this symmetry result?? why and how does it happen??


Also in my above explanation of why does the plank rotate, i said, that I apply a force on the point, the point in turn applies the same force on the mass on the left and right of it. First of all am I right in sayng so?? If I am, then if we consider the point, The masses on the left and right of this point will in turn apply back a reactionary force on the point of same magnitude which will be directed towards me(newton's third law).
so now if we consider the point, the net force on it will be directed towards me since I apply F neutons away from me, and this point recieves two reactionary forces directed towards me of the same magnitude F.
So it should accelerate towards me, not away from me, which is a contradiction. Where did I go wrong??

Thanks,
Shikhin

ANSWER: Hello again Shikhin,

I find your microscopic reasoning quite correct, i.e. the way your force is distributed from the action point to neighboring atoms. There are some mistakes in the concepts of classical mechanics, which I'll try to clear up.

The laws of mechanics teach us that if you apply force away (distance X) from the mass center COM, you have to analyze the problem as doing two things at the same time: 1) applying the force F in the center of mass and 2) applying a force-pair moment M = F*X. While 1) has purely linear transport effect, 2) has purely rotational effect. You understand 1) well, let's focus on 2). If the plank is not attached to anything, it must start to rotate about a plank's "proper axis". (At the same time its COM accelerates linearly, but that's part 1) of this problem.)

First property of proper axis is that it passes through the mass center (your COM), not through -X point. Secondly, the direction of the axis must be perpendicular to the common plane of the vector F and the connector of point X and COM (commonly the length X of the connector is called the "beam" of the force-pair moment). Take time to draw yourself a picture. Bear in mind that force F and force-pair moment M are completely different quantities, even though they occur at the same time. One linearly pushes the object, the other only makes it rotate about COM.

The Newton's 3rd law (N3L) is perhaps the most misunderstood concept in general physics. So:

[Q]Also in my above explanation of why does the plank rotate, i said, that I apply a force on the point, the point in turn applies the same force on the mass on the left and right of it. First of all am I right in sayng so??[\Q]
Yes. This is not the case of N3L, though.

[Q] The masses on the left and right of this point will in turn apply back a reactionary force on the point of same magnitude which will be directed towards me(newton's third law). so now if we consider the point, the net force on it will be directed towards me since I apply F neutons away from me, and this point recieves two reactionary forces directed towards me of the same magnitude F.[\Q]
Yes.

[Q] So it should accelerate towards me, not away from me, which is a contradiction. Where did I go wrong?? [\Q]
The point X of the plank does not accelerate towards you, because YOU are in its way! But it does transfer the reaction force on you just like its neighbors translated it onto him. What astronauts observe is that at the same time, when they push the plank (and it starts to rotate and fly away) the PLANK PUSHES THEM BACK and immediately THEY start to rotate and fly backwards! If the plank is light compared to their body mass, they may not observe the effect except after a very long time. If the plank is heavy and they are not solidly attached to the ISS, they will observe the effect of the reaction force immediately. The reaction force does not disappear if they first attach themselves firmly to ISS, only instead of them alone (inertial mass m1) the force gets translated onto the whole ISS (inertial mass m0) and the whole ISS+you (m1+m0) gets accelerated - by small amount due to mass ratio, but measurably.

I hope this helped:-).
Cheers,
Daniel





---------- FOLLOW-UP ----------

QUESTION: Hi Dan,

Thank you for the answer.
Suppose I apply a force F on the point. The point applies back a force F on my finger.
Also the plank has two reactionary forces F. One of these, cancels out the force i gave to the point. The other one is left which tries to rotate the point backwards but my finger comes in the way and so this force F is again transmitted to my finger.
So adding this one F and the above F which the point applied on my finger as soon as i touched it, the net force on my finger should be 2F???

I know it's just some silly mistake I;m making about "which" force acts "on what", but it's a bit confusing.

Also you said, that no matter where you apply the force, if the thing isn't attached to some axis, it always rotates about the axis passing from COM. And the torque of the force is then F x r.
I understand that but Torque itself isn't something fundamental like the newtons law of F=ma.
It's just a mathematical way of representing the situation.
So isn't my molecular explanation the reason for the torques??
Isn't it just that when the force isn't sufficiently big to break the electromagnetic bonds, the thing follows the mechanism i explained if the force acts on a "part" of the entire object and not the "whole" object at once??
Aren't Torques just a nice mathematical way of expressing this very thing??

Also can it be derived through the transmission of force mechanism that the thing will always rotate about the COM??

Thanks again,
Shikhin

please take your time......!!!

ANSWER: Hi Shikhin,

today there wasn't much problem answering your questions in lengthy ways, at work I just happened to have less to do today. Scarce event, but so it was. When I cannot answer, I'll delay.

Indeed there is some confusion here as to on what which force acts. It should be clearer, if you draw a picture and make a colorful distinction between a) force acting on plank b) torque acting on plank c) force acting on you and d) torque acting on you. You can set d) to zero, because rotation of the astronaut is not making the problem any more interesting that it already is. This is best to go through with a tutor, I think that when I was 15 and encountered these considerations for the first time, it was unclear to me in very much the same way. There is no justification for 2F force, remember never to add up forces, which act on different objects.

Torque exists completely independently from atomic nature of matter.  Also it is not just a convenience: it is a concept equally important as force and in essence independent. If you define a universe made entirely of discs (or balls in 3D) with axis (center) fixed in space, then all interactions between these will be only via torque: there will be no linear motion possible, only rotational. If all the discs (balls) have the same radius, there will be no principal need to define force at all! The fact that torque = force-pair moment does not make force more fundamental- it's only that in human consciousness linear motion and hence forces have slightly larger impact than rotations and torques. Nevertheless, the rotational moment J - the equivalent of mass m in rotational equations - is an important quantity of its own right.

The cohesive atomic structure of the plank is giving the reason, why the object moves as a unit and doesn't disperse. The point X you are pushing at would like to move straight, but is impeded by the inertia of all the other atoms. As X<>COM, the impeding inertia is unbalanced and hence the more massive side lags behind, which demonstrates itself globally as rotation. Perhaps this is exactly, what you meant. But beware, if no atoms existed, the massive continuum would behave exactly the same.

I'll get to the next questions during the next few days.

Cheers,
Daniel


---------- FOLLOW-UP ----------

QUESTION: Hi Dan,

I'm still not satisfied with what you said, that torques are an independent quantity and are no less fundamental than forces.

You say that torques exist completely independently of the atomic nature of matter.
Also torque is said to be " the tendency of forces to rotate things " or in other words a "twist". But this tendency or this twist arises only  from the atomic nature and the mechanism I explained??
You also said that even if no atoms existed, the massive continnum would behave the same way.
I don't understand why it would because then there would have been no electromagnetic bonds present and the point on which we would have applied theforce would have simply "broken" apart from the plank and flown away from us in a straight line.

So it is the atomic structure that is making things behave in this way. It's only these bonds and atoms that are creating a rotational effect in the thing.

So the force hasn't really got anything to do with rotating anything. But it's the atomic structure and the mechanism we discussed that makes the thing rotate.

Also, I wanted to know, that why do things always rotate about the axis passing through the COM???

When the point transfers the force to the left part which has more mass, the force is transmitted to the entire mass on the left. Why does this force vanish at points in the axis which as you said passes through the COM, and why do points on the other side of the axis accelerate in the opposite direction???

Why isn't it that the thing rotates about an xis passing through the other 'end' of the plank, i.e. in our situation , the left end, since the entire mass on the left of the application point experiences the same force forward, but it's just that it's acceleration is less than that of the right mass.

Also, according to the mechanism we discussed, there should be no linear motion, only this rotational motion, and that too about the axis passing from the other end of the square plank.... ???

Thanks
shikhin

Answer
Shikhin,

firstly, you need to first properly study the implications of "non-atomic Universe". Our Universe has atoms, you take it as a given fact and that limits your outlook. Yes, the atoms in our Universe have a means of forming bonds and this is what gives matter the tendency to stay together. But there is no reason, why not to define a torque acting on one atom - it actually exists and the reasons for the atom to respond like a rigid body is NOT AT ALL due to electromagnetic forces. It is rather due to electronic spin-spin and spin-orbit interactions, which you will study in quantum theory of matter. Similarly one has no right saying like you did "the point on which we would have applied the force would have simply "broken" apart from the plank and flown away from us in a straight line" - in non-atomic Universe there can be thousand and one OTHER ways to hold things together.

That said, I insist that I can DEFINE a working hypothetical Universe without atomic structure and yet with forces, torque and matter not flying apart under their moderate action. Last time I have modeled (the fixed rotating discs or spheres) a Universe, where Torque would be the main observable and Force would be an abstract construction without any direct impact on the objects in the Universe other than via Torque. And as this Universe can be DEFINED, the Torque definition must be completely independent of atomic or non-atomic character of matter.

>why do things always rotate about the axis passing through the COM?
Only objects that are completely free to move (!) rotate about an axis that passes through the COM. I think that the only honest answer to any such "WHY" question is that simply our Universe is such. Quite generally, science does not really answer "why" questions, it answers merely the "how" questions. My "how" answer would suggest that our Universe likes symmetry in things and a free rotation about a non-COM axis would be asymmetrical and you wouldn't be able to make the mass-point model correspondence. Observe: As you may know, we frequently model the motion of solid bodies as just motion of massive points (i.e. points positioned in COM and carrying all the mass of the body). Massive points cannot rotate, because they have mass but no size in any dimension. Massive points can ORBIT another point in space (at a distance distinctly larger than zero), but only if they move at a steady speed and there is a radial force towards that other point, tangential to the first point's trajectory. If you have a massive body rotate about a non-COM axis, the model would suggest that there is an attractive radial force pulling COM to a point on the non-COM axis of rotation. There is no such force observed, all the interatomic cohesive forces cancel out, their resulting force acting on the body as a whole is zero. So, the symmetry of the Universe would be broken if non-COM axes were allowed. In such case you would also be hard pressed to find which on of the infinite number of non-COM axes gets chosen to be at the center...

>Why does this force vanish at points in the axis which as you said passes through the COM, and why do points on the other side of the axis accelerate in the opposite direction?
It is commonly proven in the first semester of college physics that a motion of a solid body can be always decomposed into 1) rotation of the body about some COM axis and 2) motion of COM with respect to inertial frames. The acceleration of points is "opposite" because the centrifugal force is a CENTRAL FORCE - its direction is always in the direction of a central point and so on the other side of the center the force goes the other way. It also implies that the rotational center the force is zero.

>Also, according to the mechanism we discussed, there should be no linear motion, only this rotational motion, and that too about the axis passing from the other end of the square plank.... ???
No. In one of my previous messages I wrote:
[QUOTE]The laws of mechanics teach us that if you apply force away (distance X) from the mass center COM, you have to analyze the problem as doing two things at the same time: 1) applying the force F in the center of mass and 2) applying a force-pair moment M = F*X. While 1) has purely linear transport effect, 2) has purely rotational effect. You understand 1) well, let's focus on 2).[/QUOTE]
This question of yours shows that I was wrong: you actually do not understand "1)" or else, do not understand this quoted statement. Should I have written explicitly that "1)" is the linear motion you are now asking for? I thought I had been clear, but perhaps not. NOw, try to think about the problem decomposition into 1) and 2). I suggest you solve some problems, that's the best way to get used to how things work.

I will get to your next questions over the weekend, I promise.

Cheers,
Daniel