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# Re: rolling

• From: "Carl E. Mungan" <mungan@USNA.EDU>
• Date: Thu, 5 Feb 2004 11:30:42 -0500

OOPS! I'm highly embarrassed by having pushed the send button a bit
too fast. I wrote:

The *real* work you have to do *is* double because the hollow
cylinder ends up with equal amounts of translational and rotational
kinetic energy.
But the *pseudowork* you have to do is only half the real work
because it only considers the translation of the center of mass.
Friction does the other half of the pseudowork of getting the
cylinder moving.

That is, if you want to know how tired you're going to get when you
If you want to know how fast the cylinder is going to end up moving,

Let's try that again. Cylinder has mass M, radius R, and moment of
inertia gMR^2. Tension in string coming off the top of the cylinder
is T, static friction (assumed to point forward) at the bottom of the
cylinder is f. Cylinder translates forward a distance x, ending up
with translational kinetic energy K.

forces => T + f = Ma
torques about CM => T - f = gMa

simultaneous solution => 2T = (1+g)*Ma and 2f = (1-g)*Ma

real work by T = 2Tx
real work by f = 0
net real work = 2Tx (all done by you in pulling on string)
net real work = (1+g)*Ma*x using dynamics simultaneous solution
translational + rotational KE = (1+g)*K

The last two results are equal by the rules of translational kinematics.

pseudowork by T = Tx
pseudowork by f = fx
net pseudowork = (T+f)*x (only partly done by you)
net pseudowork = Ma*x using dynamics simultaneous solution
translational KE = K

The last two results are equal by the rules of translational kinematics.

SPECIAL CASE: hollow cylinder (g=1)
T = Ma, f = 0 (surprising when you first see this! strike out my
original comment about friction doing the other half of the
pseudowork!)
translational KE = rotational KE = pseudowork (all done by you) =
half of real work (all done by you)
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu http://usna.edu/Users/physics/mungan/