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Re: [Phys-L] the sign of g ... or the /direction/ of g



I think this is incredibly important to tackle head-on, and so I also am
very explicit with using vectors from day one in physics (whether algebra
or calculus based) and insist as John does that we have to talk here about
direction and not sign. Unfortunately the most intransigent students on
this point probably have had -9.8 drummed into their heads by well-meaning
but shortsighted high school science teachers, and that is always something
that is a challenge to overcome. I have had reasonable success, though, by
introducing coordinate systems and vector notation from the very first day,
and doing lots and lots of practice problems in the early sessions with
wonky orientations of the coordinate axes. Eventually most (never all) come
around to the idea that the signs simply indicate direction relative to the
coordinate axes chosen.

Todd

On Mon, May 9, 2016 at 11:37 AM, John Denker <jsd@av8n.com> wrote:

On 05/09/2016 08:28 AM, stefan jeglinski wrote:

This slays large populations of students on that first exam, and
seems to be a serious mental block for them

Suggestion #1: Don't talk about the «sign» of g. Instead
talk about the /direction/ of g. It's a much better concept.

Suggestion #2: Get them to draw the picture:
-- What is the direction of g? Downward you say?
Draw a little arrow.
-- What is the direction of increasing z?
Draw a little arrow.

=================
In more detail: Let's work backwards.

I recognize there is a difference between the pedagogical
starting point and the ending point. We can't teach
everything at once.

However, to get some perspective, let's start from the
sophisticated approach and see what it says about the
simple approach. Eventually we want everybody to think
of g as a *vector* in 3D. It has a direction and magnitude.
The direction is downward. Actually I would say g defines
what we mean by downward (not the other way around).

In contrast, g doesn't have a «sign» at all. Sign means
>0 or <0, but in more than one dimension, there is no
such comparison operator.

In the introductory course, we typically de-emphasize the
direction-and-magnitude representation. For example, speed
is the magnitude of velocity, which is absolutely correct,
but we typically emphasize velocity.

The magnitude of g is |g|. That is always positive, but
it's not interesting and not where the emphasis should be.

Many will inadvertently (or with intent) define a coordinate system
(e.g., up is positive),

That's a perfectly reasonable thing to intend. When I
see such things I assume they are intentional. I do not
discourage them. The idea that you can use any coordinate
system you like is a Big Deal, and it's never too early to
support this idea.

Technically they are defining a /vector basis/ not just a
coordinate system. This distinction is a Big Deal in polar
coordinates. The position-vector [r, θ, φ] is very different
from the vector basis {dr, dθ, dφ} at that point.

dr is a vector in the direction of locally increasing r
dθ is a vector* in the direction of locally increasing θ
dφ is a vector* in the direction of locally increasing φ

* means not a unit vector

In Cartesian coordinates the distinction is less obvious,
but still significant, at the most fundamental conceptual
level:

dx is a vector in the direction of increasing x
dy is a vector in the direction of increasing y
dz is a vector in the direction of increasing z

This is a tricky point. I've seen professors at Big Name
universities screw this up royally. So it's not surprising
that students need some time to let it sink in.

There are a lot of things that people think of as scalars
that really ought to be thought of as vectors; the vector
exists as a first-class object unto itself, regardless
of what basis (if any) we choose.

If we do choose a basis, then we can write the vector in
terms of its components. In 1D there is just the one
component. The poster child for this is electric current:
The following two currents are very different:

10 A
-------------<---------------


10 A
------------->---------------


where ->- represents the basis vector.

[they[ say "well g is always -9.8"

The comeback is: I say g is always /downward/. If dz
(i.e. the direction of increasing z) is also downward,
then the g vector will be some positive number times dz.

In 1D the distinction between vectors and scalars is
not so clear. Indeed in 1D there is a one-to-one
correspondence between vectors and scalars.

However, we can't restrict attention to 1D kinematics
forever. Eventually students will have to live in the
3D world. (Maybe 4D, but let's not go there right now.)
Still, even in 1D, there are tremendous advantages to
visualizing g as an /arrow/ with a direction. Ditto
for electric current, and a lot of other things.
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--
Todd K. Pedlar
Associate Professor of Physics
Luther College, Decorah, IA
pedlto01@luther.edu