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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:a
Some students and I in my AP C-level high school course are trying
levelproblem from the 1994 mechanics test. A ball, rolling along a
troublesurface, encounters an incline. In the first case, we have no
Then,calculating the velocity of the ball at the top of the incline.
itthe
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
Thanks.did
pure rolling up the incline in the first part of the problem?
when it
If the incline is frictionless, the ball will slide up the incline
instead of rolling. It will continue rotating at the rate it had
reached the bottom of the incline. At the top, if it gets there, itwith
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
friction, the ball rolls without slipping, and that the transitionfrom
level to incline is sufficiently smooth that no energy is lost inthe
transition. [You have to be careful about such things in this forum,and
no doubt somebody will find another assumption that I haven'tlisted].
--
Maurice Barnhill, mvb@udel.edu
http://www.physics.udel.edu/~barnhill/
Physics Dept., University of Delaware, Newark, DE 19716