Re: Rolling AP Problem

[Leigh Palmer <palmer@sfu.ca>]

> Some students and I in my AP C-level high school course are trying a
> problem from the 1994 mechanics test.  A ball, rolling along a level
> surface, encounters an incline.  In the first case, we have no trouble
> calculating the velocity of the ball at the top of the incline.  Then, the
> question asks how fast the ball would be going if the incline was
> frictionless.  Would it be faster, slower, or the same speed as if it did
> pure rolling up the incline in the first part of the problem?  Thanks.

This is a multilevel problem. I like it because no matter how I
look at it I can find something wrong with it. Such problems are
great stimulants to interested involvement among good students.

Leaving aside the question of how the transition from horizontal to
inclined motion is achieved for the moment we will extract the
conventional answer to this question. That will assume conservation
of energy, in which case the entire kinetic energy of the ball from
the bottom of the incline (0.9 m v^2) will be converted to
gravitational potential energy with pure rolling , while only the
translational kinetic energy (0.5 m v^2) will be converted if the
surface is ideally frictionless and the ball continues to rotate at
the highest point of its motion. Thus the ball will rise only 5/9
as far without friction.

There is a problem with this if the transition to the incline is
abrupt, a dihedral between a horizontal and an inclined plane. In
that case the up-plane component of linear momentum will be
conserved, with a consequent loss of kinetic energy in the bump.
Angular momentum about the point where the center of the ball is
at the instant of transition is also conserved because the external
forces acting on the ball during the transition (gravity and the
normal forces exerted by both planes) all act through that point
and thus exert no net external torque on the ball. The result of
this refinement will be different in the two cases considered. One
difference is that immediately after the transition event neither
ball will be rolling without slipping.

I'll leave the numerical calculation to you! Do it before you talk
with your students about it. The frictional problem does not have a
simple answer because specification of the problem is incomplete
without further specification, for instance of a coefficient of
kinetic friction.

But, on the other hand, the problem as it was stated originally did
not have a well defined answer either unless the coefficient of
friction exceeded a critical value...

& on & on & on... See what I mean? The discussion should under no
circumstance be allowed to end without bringing up the problem of
the slowing of a bowling ball which initially slides along the
alley (excuse my ancient terminology) and then starts to roll.

Leigh