What is the relationship between constant speed rolling motion and friction?

[flasherffff]

hi ,this is my first post (and its not HW)
i currently study dynamics of rigid bodies
using (Hibbeler)

and there's one thing about this type of motion (rolling) that just seems to elude me

for any planar motion we have 3 equations
$$\Sigma F_{{x}}={\it ma}_{{g_{{x}}}}$$
$$\Sigma F_{{y}}={\it ma}_{{g_{{y}}}}$$
$$\Sigma M_{{g}}=i_{{g}}\alpha$$

if i have a wheel rolling on a surface with constant velocity ,say a car wheel not connected to the engine

by kinematics $$a_{{g}}=\alpha\,r$$ then $$\alpha=0$$

then there is a force P acting on the mass center G
then by the first equation if a_g=0
then the friction force f must be equel and opposite of p and acting at the contact point A

but then by eq3 the moment about G causes an increase in $$\alpha$$
contradicting the kinematics equation

http://img5.imageshack.us/img5/4869/scanpic0001m.jpg

now ,obviously i got something wrong but what

 

[Doc Al]

I don't understand. You say constant velocity, yet you have some force P acting? Where does that come from?

 

[flasherffff]

i assume there is some rolling friction beforehand

 

[K^2]

If there is a force acting on the center of the wheel, one or both of the two statements is true.

1) The car is accelerating/decelerating.
2) The car is applying torque to the wheel via brakes/engine.

 

[flasherffff]

but if i don't apply a force
the rolling motion would stop due to friction

 

[Doc Al]

but if i don't apply a force
the rolling motion would stop due to friction

What friction? Are you considering rolling friction? First get the ideal case straight--perfectly rigid wheel with no friction to worry about.

 

[flasherffff]

why a perfectly rigid wheel would experience no friction
i mean static friction at the point of contact

 

[Doc Al]

why a perfectly rigid wheel would experience no friction
i mean static friction

Unless the wheel/surface deform--the source of dissipative rolling friction which stops real wheels from rolling indefinitely--there's no friction to worry about. No static friction is required to maintain the motion--it's constant velocity. The static friction is zero if the wheel isn't accelerating.

 

[flasherffff]

thanks ,i understand what i got wrong
i assumed there is a constant friction resisting to rolling just like the kinetic friction for sliding

 

[K^2]

You start with alpha=0 which means no acceleration, which is true only if there is no friction or if there is a torque applied to counter friction.