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Re: Rolling AP Problem



I'm afraid that I really don't understand the answer given below!

It seems to me that the problem is simply one of conservation of energy.

A ball that "Rolls without Slipping" has two parts to its total
kinetic energy when moving along a horizontal surface:
(1/2) m v^2 <=> translational kinetic energy
(1/2) I w^2 <=> rotational kinetic energy ("w" is omega)
Since the ball "Rolls without Slipping" there is NO ENERGY LOST TO
FRICTION because there is ZERO MOTION of the instantaneous point of
contact in the direction of the frictional force (leading to a zero work
integral).

When the ball encounters an incline and assuming it still "Rolls without
Slipping", it goes up the incline with NO LOSS TO FRICTION, transferring
ALL of both its translational and rotational kinetic energies directly to
gravitational potential energy. Since TOP is by definition the "highest
point" to which it rises, it must get to a TOP. And, I maintain that this
is the highest TOP to which it may rise.

If on the other hand, when the ball gets to the incline, it "Slips without
Rolling" up the hill, it can only convert its translational kinetic energy
to gravitational potential energy. The rotational kinetic energy is, if
you will, "trapped" due to the lack of friction to couple it to the motion
of the ball. It should, as previously described, rise to a maximum height
(which MUST BE LESS than that of the "Rolls w/o Slipping" case) and sit
spinning at this lower TOP.


On Mon, 6 Apr 1998, Maurice Barnhill wrote:

Ron Curtin wrote:

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.

If the incline is frictionless, the ball will slide up the incline
instead of rolling. It will continue rotating at the rate it had when it
reached the bottom of the incline. At the top, if it gets there, it
will be rotating faster, and its center of mass will have a lower
velocity, than when it rolled up.

I am assuming that on the level surface and going up the incline with
friction, the ball rolls without slipping, and that the transition from
level to incline is sufficiently smooth that no energy is lost in the
transition. [You have to be careful about such things in this forum, and
no doubt somebody will find another assumption that I haven't listed].

--
Maurice Barnhill, mvb@udel.edu
http://www.physics.udel.edu/~barnhill/
Physics Dept., University of Delaware, Newark, DE 19716



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