Re: [Phys-l] Equations (causal relationship)

["Ken Caviness" <caviness@southern.edu>]

Thanks to JD for the 3 uniaxial translation stage Gedankenexperiment!  Very ingenious.

I must admit that I was quite unhappy with his previous examples, because it seemed to me that he was just muddying the waters with things like (from a previous message, 5/1/06, 10:24am):

"The car accelerates relative to the table, and the table accelerates relative to the floor.  The acceleration vectors add."


This bothered me on two levels:  

(1) In intro- & general-level physics classes we have to constantly hammer the point that in constructing a force diagram, you only include forces acting on the particular object whose acceleration you are considering.  

Now in General Physics the class already knows how to add vectors, both graphically and using components.  We already practiced that in adding relative velocities, for example:  The wind is blowing at a certain velocity v1_vector, a plane flies at velocity v2_vector, find the velocity of the plane relative to the ground.  I don't recall off-hand doing a similar example using relative accelerations, but I'm sure my students would consider it routine, having successfully added relative velocities.

_BUT_ notice that adding relative accelerations seems to be quite different (at least at first glance) from adding up forces.  All my forces are acting on the same object, whereas the accelerations are relative as in the quote above:  the acceleration of the car relative to the table, the acceleration of the table relative to the floor.  Again, at first glance this appears quite different, separate accelerations due to separate forces acting on _different_ objects.  Some force acts on the table, moving it across the floor, some force acts on the car, moving it across the table.  Yet it seems obvious that the vector addition of accelerations means that there are forces which could be added, too.  A key realization is that if the car weren't accelerating relative to the table, it would be because it was subjected to a force of the same magnitude and direction as that to which the table is subject, in this case a friction force acting on the car.  So in short, we can add the relative accelerations OR we can add all the forces acting on the car.  It doesn't matter.  My first objection down.

(2) In special relativity F = gamma m a, still for constant mass cases.  Our naïve vector addition for velocities and for accelerations fails, whereas we can still make force diagrams and find the net force acting on an object.  How do we resolve that?

Must go back to preparing finals now.  I wish I had time to follow this discussion more closely.

Ken Caviness
Physics Department
Southern Adventist University


-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Tuesday, May 02, 2006 5:05 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Equations (causal relationship)

Edmiston, Mike wrote:
> > Why don't we just agree as follows:
> >  a) Sometimes ma is calculated from F and sometimes vice versa;
> >  b) The equation F=ma covers both cases.
>
>
> Absolutely.  I've never had a problem with that.

We should probably just leave it at that.

==================

But being a glutton for punishment today:

> I can imagine a particle having its net acceleration resolved into a 3D
> coordinate system.  I personally wouldn't say that a(x), a(y), and a(z)
> are three simultaneous accelerations; rather, I would say they are
> components of the one acceleration.  Then, when a(net) is zero, I would
> say all the components are zero.

Assuming x y and z are orthogonal, the only way the vector a(net)
can be zero is for each of its components to be zero.  I didn't
think that issue was in dispute.  I thought the topic of recent
discussion was three nonzero linearly-dependent accelerations that
added to zero.

> Where I have difficulty is imagining three simultaneous nonzero
> accelerations that add to zero.  The only observable result is a(net)
> equals zero.  I cannot oberve the individual nonzero accelerations.  I
> don't think it is a failure of my imagination or my experimental skill
> that I can't figure out a way to measure them, I think they cannot be
> measured in principle.

I think they can ... in principle and in practice.

Here's how I visualize it:

Imagine three uniaxial translation stages, of the kind you see all
over the place on optical tables.  Stack three of them together,
one atop the other.  The particle of interest is attached to the
top of the stack.

Arrange that the first stage has its axis aligned with force F1, the
second one aligned with force F2, and  the third one aligned with
force F3.

Arrange that the first one accelerates at the rate F1/m, the second
one at the rate F2/m, and the third one at the rate F3/m.

The three accelerations are nonzero.  They add to zero.  You can
see the stages moving, even though the particle does not move.

This is not the only combination of accelerations that adds up
to zero ... but F1+F2+F3 is not the only combination of forces
that results in equilibrium.

Anything you can say about force vectors I can say about acceleration
vectors.  They're vectors;  they add tip-to-tail.