I should probably leave well enough alone. But oh well. How can I
resist throwing out one more example of work-energy which I'm sure
will provoke others? I only thought of it today because we were doing
rolling problem in class.
A sphere rolls down an incline without slipping. The rolling arises
because of static friction pointing up the incline. Does this
frictional force do work?
Answer 1: Yes. When you dot it with the ball's displacement, you get
a negative answer. This explains why the ball takes longer longer to
get down the incline than it would if it slipped frictionessly.
Answer 2: No. The contact point between the sphere and incline does
not move, because the sphere does not slip. This explains why the
sphere does not warm up.
I hope, even though I am reasonably certain that it is a vain hope,
that you will not be surprised that I agree with *both* answers (in
two different teaching contexts of course).
From the point of view of the work-energy theorem here are the
corresponding analyses:
1: W_nc = -f*d = delta (E_mech) = 0.5*m*v^2 - mgh
Here the rotations of the particles about the cm is considered a form
of internal energy. All we know about mechanical energy is what we
learned in the work and energy chapters: translational KE,
gravitational PE, elastic PE.
2: W_nc = 0 = delta (E_mech) = 0.5*m*v^2 + 0.5*I*w^2 - mgh
Here we decided to redefine (extend) our notion of the kinetic energy
of a system to include bulk rotational energy.
It is left as an exercise to the reader to verify that f*d =
0.5*I*w^2 so that the two analyses are equivalent.
You may not *like* one of the two analyses above. I on the other hand
happen to *like* both and think students can benefit from both.
To each his own, Carl
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu http://physics.usna.edu/physics/faculty/mungan/