Rolling and Friction: Explaining the Paradox

[Chen]

Suppose there's a wheel on the ground, radius R, and I'm pulling it with a string connected the top of the wheel with force T.

We know that if we want the wheel to roll, there has to be friction between the ground and the wheel. But at the same time we ignore that friction and say that the only force on the wheel is T, and the torque is:
N = RxT

So why do we require the existence of friction but also sort of ignore it? I know that it's correct I just want to understand why.

Thanks.

 

[brewnog]

I'm not quite sure how you want the string to be attached, but to answer your question, we ignore friction so that you can neglect any minor losses which are due to it, but you need to know it does exist otherwise the wheel might slide before it rolls, and this isn't how you want to approach your problem.

 

[nishant]

here is the answer

you should appraoch the problem like this:
if friction is not there your sphere will slide because there is no force to ratate it about the centroidal axis.
here the friction that comes into effect is static friction which by it's defination does not do any work. That is the reason why we do not take friction into account while solving the motion's eqation.

 

[krab]

The friction acts on the wheel to start it turning. You can calculate the force of friction from the acceleration of the axis and the moment of inertia of the wheel. This will definitely be a contribution to the force you called T. If you are pulling a wheel by a string, the force needed to start the wheel spinning is 1/3 the total force. This is the friction force and it definitely is not ignorable. In a sense, the static friction sets up the proportion of energy that ends up in rotational (Iomega^2/2) vs. translational (mv^2/2) (v here is the velocity of the c.of m.).

OTOH, if you are pulling a large vehicle where the mass is much larger than the mass of the wheels, you can often safely ignore this effect.

 

[Doc Al]

krab already answered, but I'll add my two cents.

We know that if we want the wheel to roll, there has to be friction between the ground and the wheel. But at the same time we ignore that friction and say that the only force on the wheel is T, and the torque is:
N = RxT

Cleary if a frictional force exists, you can't ignore it! The torque would not be RxT!

krab figured out the frictional force needed to have the wheel roll without slipping (F = T/3). In doing so, he treated the wheel as a cylinder or disk ($I = 1/2mR^2$).

Just for fun, figure out the frictional force needed if you model the wheel as a ring ($I = mR^2$).

 

[spacetime]

If one lifts the wheel off the ground and then pulls the top part of the wheel
with a force T, it'll certainly rotate.
So how is this situation different from the wheel being on the ground+friction absent?



spacetime
www.geocities.com/physics_all/

 

[Doc Al]

If one lifts the wheel off the ground and then pulls the top part of the wheel
with a force T, it'll certainly rotate.
So how is this situation different from the wheel being on the ground+friction absent?

It's not different. If a friction force exists, it contributes to the net torque about the center of mass. The static friction simply prevents slipping between the wheel and the ground.