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Re: [Phys-L] treating force as a vector ... consistently

Let me start by saying that, aided and abetted by the FCI, some of us (me,
for example) make a fetish out of the 3rd law. But let's continue...

As mathematical entities, the force the table exerts up on the book and the
force the book exerts up on the Earth are equal in magnitude and in
direction. So I agree, they are two equal vector quantities. But JD, are
you saying that makes them the same thing? I think they are two different
things that happen to be equal, this time, because the book is in
equilibrium.

We have discussed here before the idea that a vector does not have a "point
of application" as part of its mathematical definition. But somewhere in
our definition of forces, we have to build in (at least implicitly) the
idea that a force is something that is applied to an object and that a
given force can only affect the object it is applied to -- though
obviously, its 3rd law counterpart can affect the "other" object.

I guess I need more clarification of your point above:

"My point is that F_b and F_c are two names for the same thing. They
are the *same vector* since they have the same magnitude and direction."

I'd say that two force vectors can be equal and yet distinct. Not even
just because they act on different objects as these two do. What about two
vertical ropes symmetrically holding up a weight?



On Sat, Aug 20, 2016 at 12:50 PM, John Denker <jsd@av8n.com> wrote:

On 08/20/2016 07:47 AM, Philip Keller wrote:

Second, the phrase "equal and opposite" shows up in two distinct contexts
that are easy to confuse.

Yes indeed.
-- equal and opposite as required by the third law, always;
-- equal and opposite as required by the second law, if-and-when
the object is in equilibrium.

NOTE: For clarity, in what follows, I assume everything is in equilibrium.

I do as questions like this on multiple choice tests. But I warn
students
in advance and then again on the test itself, in the text of the item: I
am
not asking you to identify an equilibrant but rather a "reaction" force
as
required by the third law.

Now there's a problem.

There is a force-ish thing "here" that we know based on the third
law, and there is a force-ish thing "over there" that we know based
on equilibrium arguments.

From a physics point of view, these are two different things, i.e.
two different /dynamical interactions/ ... but from a mathematical
point of view they are the same vector. The math is unambiguous
and uncompromising: a vector has a direction and a magnitude, but
it does not have a location. Talking about a vector "here" or a
vector "over there" does not make sense. It is a distinction without
a difference.

Let's be clear: It makes no sense to tell students that force is
a vector and then turn around and ask them to distinguish a force
"here" from a force "over there". It is a crime against the laws
of mathematics.

It's no wonder the students are confused. In the introductory
course, they probably won't be able to put their finger on it
and explain to you why they are confused, because they don't
have a firm enough grasp of the mathematical formalism ... but
it's not their job to explain stuff to you.

====

To say the same thing another way: One can reasonably ask students
to /show the work/ and to explain their thought processes. Consider
the following:

*) Force F_b must be equal-and-opposite to force F_a.
Rationale: second law, plus the fact that the system is in equilibrium.

*) Force F_c must be equal-and-opposite to force F_a.
Rationale: third law, or (equivalently) conservation of momentum.

My point is that F_b and F_c are two names for the same thing. They
are the *same vector* since they have the same magnitude and direction.
The main thing that is different is the method of calculation.

THEREFORE: Suggestion #1: Structure the question to ask about the
method of calculation (not about the vector itself).

The book has mass m. It rests on the table, in equilibrium.
The table exerts a contact force F_x on the book.
Choose the value of F_x and the /most direct/ explanation:
A) F_x is the same as mg, as required by the second law.
B) F_x is equal-and-opposite to mg, as required by the second law.
C) F_x is the same as mg, as required by the third law.
D) F_x is equal-and-opposite to mg, as required by the third law.

As a rule, in physics, in business management, and in general:
/Measure the thing you care about/

AFAICT we are talking about assessing /how the students know/ the
force. That's a reasonable thing to assess. Design the question
to focus on the thing you care about!

Beware that in realistic situations, there are generally multiple ways of
figuring stuff out. (Many schemes for /checking the work/ are predicated
on this.) Therefore questions about "how do you know" tend to have more
than one correct answer. Barometers and all that:
http://www.snopes.com/college/exam/barometer.asp

Suggestion #2: Teach students to translate third-law questions into the
language of momentum transfer and momentum conservation. That's what
smart people do. There are lots of third-law questions where /nobody/
can figure out what's going on in terms of action and reaction.

The gravitational interaction transfers downward momentum from the earth
to the book. That is synonymous with transferring upward momentum from
the book to the earth. It is tricky but possible to draw diagrams of
this:
https://www.av8n.com/physics/force-intro.htm#sec-momentum-flow

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