If a wheel is rolling without slipping

[ximath]

Dear All,

I have learned that if a wheel is rolling without slipping, then the velocity of point of contact of a wheel with respect to the ground is zero instantaneously.

However, I am having some problems while trying to understand that.

We can express the point of contact's velocity as;

V = Vrot + Vtrans

It is zero because simply Vrot = -Vtrans.

But after that instant, the point moves and an adjacent point becomes the point of contact -- thus rolling appears.

If the point's velocity is zero, then it should stay at the bottom as the point of contact. And if it stays at / as point of contact, its velocity should still be zero.

There is a circular logic here;

How does it move if its velocity is zero ? And how does it have velocity greater than zero if it doesn't move ? ( since Vrot = -Vtrans if it stays at the bottom, its velocity would be zero.)

Thanks in advance.

 

[Doc Al]

If the point's velocity is zero, then it should stay at the bottom as the point of contact. And if it stays at / as point of contact, its velocity should still be zero.

The velocity of that point (with respect to the ground) is momentarily zero. But its acceleration is not zero.

That path traced by that point is a cycloid. Check it out: http://en.wikipedia.org/wiki/Cycloid"

 

[ximath]

It must have an inward acceleration, (towards the center of the wheel) and this acceleration is what causes the rotational velocity.

But if it stays at the bottom as the point of contact and it has translational velocity, then net velocity would still be zero and particle should still have zero displacement, which is not the case ?

 

[Doc Al]

It must have an inward acceleration, (towards the center of the wheel) and this acceleration is what causes the rotational velocity.

Right. The acceleration of the point at the bottom of a wheel is momentarily upward.

But if it stays at the bottom as the point of contact and it has translational velocity, then net velocity would still be zero and particle should still have zero displacement, which is not the case ?

I must be missing your point. Realize that the part of the wheel in contact with the ground is continually changing. (See my last post about the cycloid path taken by a point on the edge of the wheel.)

 

[ximath]

I have checked cycloid out, it illustrates it very well but still,

If velocity of a point is zero, then it should not move. How does the part of the wheel in contact with the ground change if there is a point with zero velocity ?

 

[Doc Al]

If velocity of a point is zero, then it should not move. How does the part of the wheel in contact with the ground change if there is a point with zero velocity ?

Again, its velocity is only momentarily zero. Just like tossing a ball vertically. At the top of the motion its velocity is zero, yet it falls back down.

 

[ximath]

Hmm, I liked that analogy. It enlightened something.

In our case, there is an inward acceleration. That inward acceleration at the point of contact would produce a velocity that is tangent to the wheel. And the translational velocity is constant and opposite to that. So we say momentarily, V = 0. By meaning that momentarily, Vrot = - Vtrans.

Different from the ball analogy, here we have two velocities that are equal to each other which makes net velocity 0.

Hmm, I guess I got it. To illustrate it,
Let's assume Vrot = Vtrans = U in the instant V = 0. Then after a while, due to the acceleration, Vrot = U + h (h is the contribution of the acceleration in that "while") but Vtrans is still equal to U so Vrot > Vtrans and the point moves!

Am I correct ?

 

[ximath]

Hi again,

Sorry for "resurrecting" this thread, but I got an idea about rolling and friction related to this topic and wanted to make sure that it is correct.

I have just learned that friction does zero work in rolling without slipping condition. The first reasoning I have been able to come up with is; the point of contact at rest momentarily (because Vrot = Vtrans) and after some time due to the acceleration (which is inward) it stars to move, as it moves since acceleration is inward and its initial velocity is zero, the velocity caused by the acceleration that makes the particle move is pointing through the center, thus its displacement is pointing through the center for a while which is always perpendicular to friction force, thus making F dot product d = 0.

Might this be the reason behind the fact that friction does zero work on rolling without slipping condition ?

Thanks in advance.

 

[ximath]

Anyone, please ?

 

[Doc Al]

Might this be the reason behind the fact that friction does zero work on rolling without slipping condition ?

It's even simpler than that. The point of application of the force (which is always at the point of contact) does not move. Thus there is no displacement: W = F*d = F*0 = 0.

 

[ximath]

the point doesn't move for an instant but must not it move after a while, otherwise how would it roll ?

 

[Doc Al]

the point doesn't move for an instant but must not it move after a while, otherwise how would it roll ?

Sure, but as soon as it moves, friction no longer acts on it. Friction only acts on the point of contact (which constantly changes, of course), which is always at rest.

 

[ximath]

I guess I'm getting confused now... We have a point, which is in contact with the ground. And a friction is applied to it. It does not move for an instant, but after a while it must move to make another(point) point of contact. What I would suggest is to make this movement, it must move perpendicular to the direction of friction force; thus friction does zero work.

The explanation "friction does no work because displacement of that point is zero" makes me feel confused because actually that particle must have displacement in order to yield for another particle. What I tend to think is displacement is not zero

 

[Doc Al]

The displacement of the point of contact while friction acts is zero--it doesn't move. The "point of contact" is an ever-changing segment of the wheel. But friction only acts on the segment of a wheel when that segment is at the bottom and not moving with respect to the ground. Of course, when that segment does move, it loses contact with the surface and thus no friction acts on it.

A similar example is walking without slipping. Although friction propels you forward, it does no work on you, since the point of contact does not move while the friction is being applied.

 

[ximath]

Thanks a lot! That makes sense.