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Re: [Phys-l] block sliding on a skateboard



On 02/04/2012 07:41 PM, Jeffrey Schnick wrote in part:

x_s = .496 m

I agree with that, in its context.

There is (as usual) more than one way of obtaining the answer.
Some folks may find the following version amusing. I like it
because I can do it in my head.

Executive summary: rather than calculating directly how far the
block slides, calculate how far it /doesn't/ need to slide at
the end of the process. This obeys a simple scaling law, if we
make the "right" set of assumptions.

Note: All velocities etc. here are relative to the lab frame.

Movie #1: Skateboard immobile. α = 1. Distance = 1/2 a t^2.
In more detail: D[1] = 1/2 a t[1]^2.


Movie #2: Skateboard moves. α = 1.1 Distance = 1/2 a t^2
In more detail: D[α] = 1/2 a t[α]^2.
where the acceleration (a) is the same as before, and the time
t[α] is shorter than t[1] i.e. t[α] = (1/α) t[1]. The acceleration
is the same because we assume (!?!?!) that sliding friction is
independent of relative velocity. The idea that the absolute
acceleration is the same -- whether or not the skateboard is free
to move -- is counterintuitive, but that's what our assumptions
demand.

Let's pay attention to the block only. That means movie #2 is
exactly the same as movie #1, except that it ends earlier.


Movie #3: Consider the difference between the two movies, i.e.
the snippet of movie #1 that extends past the end of movie #2.
The duration of this snippet is one 11th of the total time t[1].
Since distance scales like time squared, and since the block has
zero velocity at the end of the movie, the distance traveled in
this snippet is less than D[1] by a factor of 11 squared. This
is the distance that the block /did not/ travel in movie #2.

You can check that .496 is equal to .500 * (1 - 1/121).

More generally, introduce the parameter β := (1 - 1/α) i.e. the
mass of the block divided by total mass of the system. For the
specific example we are considering, β = 1/11. We have found
another scaling law: The distance that the block /not/ have to
slide scales like β squared. The time involved scales like β
to the first power.

This new scaling law has virtually no practical significance, because
it strongly depends on the assumed friction/velocity relationship.
This makes it much less interesting than the less-restricted less-
dependent scaling laws discussed in the previous email. The assumption
that friction is independent of velocity is probably never accurate at
the 1% level or even the 10% level. Reference:
Rod Cross
"Increase in friction force with sliding speed"
http://ajp.aapt.org/resource/1/ajpias/v73/i9/p812_s1
Furthermore, lubrication would invalidate the assumption at roughly
the 100% level.

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

On 02/04/2012 03:12 PM, LaMontagne, Bob wrote:

I will need to revisit this with my students (big mea culpa).

IMHO a /small/ mea culpa is plenty. It's still a nifty problem.

In fact it is a teachable moment. It illustrates (some of) the
difference between end-of-chapter problems and real-world problems.
In the real world, essentially *every* problem is ill-posed.

Students need to learn how to handle ill-posed problems. No amount
of practice with typical end-of-chapter problems will help with this.

The first step is always to /notice/ that the problem is ill-posed.

Huge hint #1: If group A gets answer A and group B gets answer B,
it is a sign that the problem may be underspecified, and the two
groups are making different assumptions.

Similarly, if group A thinks the problem is easy and group B thinks it
is hard or impossible, it may be that the problem is underspecified.

Once you notice that the problem is ill-posed, there are tried-and-true
ways of coping with it.
http://www.av8n.com/physics/ill-posed.htm

Students need experience dealing with ill-posed problems!

"Fixing" all the problems to make them well-posed is not good in the
long run.

======

Another teachable moment: Always look for the scaling law.

In my dreams, I dream of an introductory physics book that says "always
look for the scaling law" and explains how to go about it.

For example: If you can't find a scaling law for X, maybe there is a
nifty scaling law for (1-X). And so on.

Scaling laws are easy to use and verrry powerful. They have been central
to physics all along, from Day One of modern science (1638) to the present
day.
http://www.av8n.com/physics/scaling.htm