Re: Newton's 2nd law Lab

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Concerning where Rick wrote:

> The experiment is normally done to confirm N2, in that the weight of the
> falling mass is the applied force, the mass of the system is the mass of the
> cart plus the falling mass, and the accelerations are calculated from
> measurements of time and distance.  Plots of acceleration versus weight
> produce linear graphs such that y=mx becomes F = ma (the slope is 'm').  The
> variation on the experiment is to NOT let students know the numerical value
> of the Force.

But if the weight of the falling mass is the applied force then why
does the cart move *horizontally* and the center of mass of the system
move at downward tilted angle when N2 says that the acceleration of the
CoM should be parallel to the applied force?  Clearly, you cannot
ignore the pulley's force.  Part of F = m*a is that the directions of the
vectors on each side of the equation are the *same*.

> David objects to ignoring the Normal Force of the track or the pulley force.
> The Ealing air tracks have an 'air pulley' that lets recording tape float on
> a thin cushion of air--thus producing a nearly frictionless pulley.  The
> Normal force of the track is perpendicular to the direction of motion and
> therefore does not contribute to the acceleration of the cart.

On the contrary, the track's normal force prevents the cart from falling
through the track and thus *does* affect the direction of the motion of
the cart by eliminating any vertical component of the cart's motion by
precisely canceling against the cart's own weight (weight meaning here in
the typical introductory physics way as the downward force of the Earth's
gravitation on a body), and the resultant direction (and even acceleration
magnitude) of the cart's motion is different (i.e. horizontal) than if
that normal force were absent which would result in the cart accelerating
in a complicated way on the end of a downward swinging tape.

I don't care about the negligible tangential friction forces on the
string going over the pulley.  I care about the much stronger forces
acting radially away from the pulley's center on the string.  The
resultant of these forces has a horizontal component which is sufficient
to accelerate the cart horizontally (via its imposition of a horizontal
tension in the horizontal part of the tape) and has a vertical component
to provide the vertical tension in the vertical length of tape which
tends to retard the acceleration of the falling mass.

> This is a Lab for INTRODUCTORY PHYSICS STUDENTS.  As seems often the case, I
> think David is too concerned with here with complicated details of 'real
> world' physics.

No, I'm a theorist. I'm *not* concerned with the trivial complicated
details; I'm concerned here with the *idealized* problem and with all the
relevant conceptually important factors.  It's just that the experiment
does not seem to do what it is claimed to do regarding demonstrating or
discovering N2 when major contributions to the total force on the system
are ignored, and the parts of the system and its CoM accelerate in
directions different than the included applied force.

>                 However, this experiment, in many different, forms DOES
> produce results quite consistent with the typical modified Atwood's machine
> analysis WHICH relies on an application of Newton's Second law to analyze.
> I really don't see the problem!

Other Atwood's machine problems (even the double vertical ones) *also* do
not demonstrate N2 when they normally are solved for just the
accelerations of the masses.  To see N2 on the composite (double vertical)
system one has to show that the acceleration of the CoM of the 2 masses
times the total mass is equal to the difference between the masses' total
weights and the upward vertical force exerted by the pulley on the string.
If a composite system point of view is not taken and the separate masses
are treated individually, then the tension in the string needs to be found
so that the total forces on each individual mass can be found and
then compared with each mass's own acceleration.

The justification for keeping the total mass of the composite system
fixed involved an argument about the composite system's motion.  My claim
is that the composite motion does not describe N2 as described.  But if
the tension in the tape could be somehow determined then N2 could be
verified (or maybe even sort of discovered) for the individual motions of
each mass separately.

> My twist on the typical experiment makes the analysis a little more
> interesting than the typical "OK, we know F = ma, so let's measure F, m, and
> a and see if indeed F = ma."  The assumption that the force of the falling
> mass IS proportional to that mass does severely weaken the 'discovery' part
> of the experiment, although the students don't realize that, but that's why
> I included a ;-) at that point in the description.

To discover or verify N2 one needs to show *both* that the magnitude of
the total force F is the product of the total mass and the magnitude of
the acceleration a, *and* that the direction of F is the same as the
direction of a.  When you have an experimental setup that has different
pieces going different ways it complicates the analysis and makes this
second part much more difficult.

David Bowman
dbowman@georgetowncollege.edu