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Re: rolling

On Wed, 4 Feb 2004 10:17:01 -0500, Carl E. Mungan <mungan@USNA.EDU> wrote:

Skip wrote:

Speaking of rolling.
Some students and I were working on the problem wherein a rope is wou=
nd around a barrel to pull it forward. The rope pulls the top of the =
barrel as it unwinds. When you calculate the acceleration of the barr=
el, it is greater than would it would be if the rope simply pulled at=
the center of mass. This must imply that the friction force on the b=
arrel pushes it forward, a surprising result to us at least.
Anything wrong with our analysis?

When I think about problems involving friction, I first try to visualize
what would happen without friction. Because that is not really very "real
world", I simulate that with how can I minimize the effect of friction.

In this example, I see a barrel being pulled to the right and starting to
rotate CW. The frictional force is limited by mu and Normal, so if I pull
really hard, I can make the effect of the frictional force essentially
disappear. What happens? The barrel spins CW rapidly and starts moving to
the right.

Now lets look at the point of contact. If the barrel is spinning CW, it is
moving to the left relative to the ground. But the barrel is also moving
off to the right, which dominates?

Gut feeling is that the spinning effect will dominate in this example
because (unless you have a perfectly thin band for a barrel) the moment of
inertia < mr^2 and it is "easier" to set the thing spinning than to move
it. If the rope was wrapped around a surface with smaller radius, the
torque would be diminished and the velocity would dominate. Therefore, in
this example, the net relative velocity of the contact point is to the left
and therefore the friction force will be acting towards the right. You are
correct, the friction force should be acting in the same direction as the

In general, the kinetic friction acts in a direction that opposes the
relative motion between the two objects. The static friction (as in this
case) acts in the same direction that the kinetic friction would act if the
object started to slide.

I'm a little confused by your statement about friction and normal forces.
For no slipping, we have static friction, which is not proportional to
normal force anyway. Did you intend that the maximum static friction is
not proportional to the normal force?

Nothing wrong. See:
and references therein. It comes about because you simultaneously
have to consider forces and torques, because linear acceleration =
radius * angular acceleration for no slipping. Thus we do *not* in
general have the condition that the friction is directly proportional
to normal force and other "common" rules about friction.

This particular example is a favorite counter-example of mine to the
question: Is friction always opposite to the direction of an applied
force (in the absence of horizontal forces other than friction and
the applied force)?

This problem is also a wonderful case study in the careful
distinction between pseudowork and real work. See Chabay & Sherwood's
text. Carl
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040