Expert: Daniel Mazur - 8/18/2009

Question
QUESTION: Hi Dan,

Still wanted to discuss a little more about it.

Consider the plank again.
In free space, if I apply a force on it anywhere apart from the COM axis, it will rotate about the COM axis. The angular acceleration will be determined by the formula T=I@, where T is the torque of my force about the COM, I and @ being the moment of inertia and angular acceleration respectively.

No matter wherever I apply the force, it will rotate only about the COM axis. Why so??

Assuming that the molecular model I gave is correct, apart from the fact that the force F I apply is not transmitted equally to all the constituent particles, as we discussed in the last question, how is it that it always rotates about the COM axis??

I know the mathematical rules of considering the rotational equilibrium and translational equilibrium separately, by calculating the "torque" about the COM axis in free space and considering the force F to be acting on the COM for linear motion.

But those are just our mathematical ways of simplifying the situation, but what is it that goes on fundamentally when I apply a force F on the plank??
The basic reason as we have already discussed many times before, are the electromagnetic forces which transmit my force F to the entire plank. And as we discussed in the last question this force F is not equally transmitted to all the constituents equally, then why is it just so necessary for the thing to rotate about the COM only???  

Thanks,
Shikhin

ANSWER: Shikhin,

I don't know, if I will answer this time around, what you are trying to get at, but I think I found one basic thing that we have not talked about yet. It may help to get it out of the way, please post me a follow-up with your questions that will remain.

You have still not told me, if you what to study motion under
A) a constant F(t)=F=const (constant direction and magnitude) or
B) under constant moment M(t) = F(t) (*) X(t), where (*) stands for vector multiplication (constant magnitude of F, constant length of x, but changing directions).

In case A) the plank will rotate first until the COM with the point of acting force F lie on a line parallel with the direction of F. From then on the plank won't rotate, but will only move linearly... It will oscillate like a pendulum, I think, but due to the forward acceleration the energy of oscillations (which won't increase beyond some value) will become negligible by comparison.

In case B) the global, long-term observation will be that the plank will rotate about a COM-axis and at the same time its COM will MOVE ON A CURVE - not a line, nor a circle - in time! Why, and how does this agree with what we have discussed and what I have said thus far? The reason is that the method of separating linear and rotational motion is only valid as an infinitesimal approximation - that is for times so short that there is no practical difference between cases A) and B).

Once we get past the region of validity of this approximation (still practically very useful, take my word for it), we have to make choice, which case we want to solve our equations for. If it is B) like now, we will be obliged to solve an equation in which there is both transitional motion of COM and rotational motion of plank about a COM axis, while the force F(t) changes direction with time (so that it, for example, stays normal to the large face of the plank)! This time dependence is something we never had to deal with in the infinitesimal approach. We can break the global motion of the plank down into a series of infinitesimal steps in time instants t_i. During each step the force F(t_i) will have an effect of a tiny LINEAR motion and a tiny ROTATIONAL motion of the plank about a COM-axis. However, between the steps the direction of force F will change (!), so the tiny linear sections of subsequent steps are NOT GOING TO BE PARALLEL to each other. Instead, they will draw a polygon, which in the infinitesimal limit (delta_t = t_(i+1)-t_i -> 0) becomes a smooth curve. The center of its osculation circle does NOT lie on a COM-axis of the plank and frequently (for small distance x of the point force from COM) will not even lie anywhere near the plank either.

To sum up, in case B) we will observe a global COMPOSITE MOTION: The plank will ROTATE about some of its COM-axes and at the same time its rotational axis (a COM-axis) will MOVE ON A CURVED TRAJECTORY through space. Under the continuous force the plank keeps speeding up both parts of its motion until it reaches the region of relativistic effects, or if a real plank reaches a breaking point of its internal cohesive forces.

So, this could have moved us a bit further, I will be grateful for your feedback/follow-up.

Take care,
Daniel



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

QUESTION: Hi Daniel,

The situation B is quite interesting and I had never given a thought about what will happen if the force is always perpendicular to the face of the plank so that its direction keeps on changing.

I understood your explanation for what will happen in situation B totally.
But what I'm trying to get at hasn't got much to do with the case B.

Let's just consider a Force F that acts on the plank for a tiny little bit of time so that during this time neither the direction of the force changes, nor does the axis start moving in a curve.
Something like just a "touch" with our finger on the plank.

So now my question really would be, that when I apply the force F on some part of the plank, this force is transmitted to all the constituent particles. And as we had discussed, this force F will not be equally transmitted to all the constituents of the plank, i.e., every constituent will perhaps receive a force F'_i which would be different for every particle. And the magnitude of the transmitted force would perhaps decrease with the distance from the point where I had applied the force F.
First of all, I want to know whether this is really the case or not.

Now if it is so, we can look at it this way that every constituent receives a different force F'_i and thus moves forward with different accelerations (assuming every constituent has same mass). And so some parts of the plank have an acceleration that is comparatively less than the others and so they lag behind the others. And since we assume that the cohesive forces don't give up, the entire thing has to move forward together but different parts of it have different acceleration due to the different transmitted forces they received and thus the net effect as we observe it is rotation.
And then my question really is that
first, you said that no matter where I apply the force F (apart from any point on the COM axis itself), and if the plank is in free space, it will always rotate about the COM axis...why is it just so necessary for it to do that?? I know it would be "symmetrical" but why is it that no matter where I apply the force, the points lying on the COM axis just don't turn?? It looks like the transmitted force F' that these particles lying on the COM axis receive, is used to translate then linearly and not rotate them. Why can't it be that the axis lies a little left or right to the COM axis and the particles on COM axis also go in rotational motion like other particles since the force F' they receive is less than F but greater than what the particles lying to the left of the COM axis receive (assuming that the force I applied was on the right side of the COM axis)???
And also another contradiction that we get from our model is that suppose I applied the force somewhere in between the COM axis and An edge. And as you said that the Plank will rotate about this COM axis passing through the center, then the points lying on the edge have the maximum linear acceleration, which is a contradiction to our model since the force F I apply gets diminished as it travels in any direction from the point where I applied it, the points lying on the edge should experience a force which is less than F and the point where I applied it should experience the maximum force which is of course F, and thus it should be the point where I applied the force that should have the maximum acceleration.
I know I'm wrong in saying the above, and would like you to correct me, since even I can't picture the situation in my head as to how would a point lying in between the COM axis and the edge having an acceleration more than that of the edge and COM axis, move.

Secondly, If we are considering this force F that's just a finger touch lasting for a very small amount of time, Will the COM of the plank also go forward with an acceleration a=F/M ??  

If yes, I would really like to know the dynamics of it ....i.e. how is the force F getting transmitted to all the constituents and as to how does it result in a translational motion of the COM with an acceleration 'a' (COM accelerates only for as long as my finger is in touch) and as to how this force is transmitted so that the thing always rotates about the COM axis....

Thanks,
Shikhin


Answer
Dear Shikhin,

it is an error to think that the F' as a result of F acting away from COM diminishes from point of action towards the edges. It is experimental observation that acceleration of different points is different in such a way that some points will accelerate more than COM or point of force action X and some will accelerate less than them. What you were describing was the case of a membrane or generally a soft elastically easily deformable object, not a rigid body. Well, to some extent any real body deforms elastically a tiny bit, but then you must include the energy of the deformation into your equations of motion, because otherwise the conservation laws won't hold. Then of course you run into many complications, because the deformation changes the shape and hence COM and rotational inertia J0 of the plank, so the new rot.inertia will be dependent on the force you use.... and so on. This is far beyond what one can calculate analytically and that is where you are heading with your queries about "how the force F is actually transmitted". Everything about the separation of linear motion and rotational motion relates to the RIGID BODY. So far you have not accepted it and that's why we cannot find common ground.

[Q]If we are considering this force F that's just a finger touch lasting for a very small amount of time, Will the COM of the plank also go forward with an acceleration a=F/M ?[/Q]
RIGID BODY again:
COM is the very point (together with the points on the rotational COM-axis), whose acceleration is a = F/M !!! The force action point X has bigger acceleration than F/M, because it is
a_x = F/M + x*alpha_x = F/M + x*(F*x/J0),
i.e. it has an added term containing angular acceleration with respect to COM axis.

Why is it a COM axis? Like I already told you, start with a freely rotating RIGID BODY without any force. If the rotational axis was anything other than a COM-axis, the free rotation would result in a NON-ZERO TOTAL of all centrifugal forces acting on various bits of the body and thus global acceleration would be observed. As no acceleration is observed, it MUST be a COM-axis that the body is rotating about. It is a nonsense in our universe that just because something freely rotates at a constant angular velocity, a global acceleration would result - that's an experimental fact. If you are asking why, then again you are questioning, why our universe works the way it works and an answer to that does not lie in the realms of physics. You could try some religion for that ;-), those are the ones answering questions "Why does it work the way it does?". Physics only answers questions "How does it work?"

If you accept this, you may ask then: why the rotational axis with force applied is not different from the rotational axis in free rotation? I think that in THIS question you would already be slipping out of the infinitesimal approximation into the finite time domain, where the COM moves on a smooth curve with a finite radius of curvature. If the time domain is tiny, the COM moves on a straight line and this DIRECTLY IMPLIES that the rotational axis must go through COM! If the actual axis of rotation was out of COM, then COM itself would have to move on a curve (circle with respect to that other axis of rotation) and that contradicts the infinitesimal approximation.

I hope I have cleared things up another bit.

Cheers,
Daniel