Rotational Mechanics:Small Problems

[Doc Al]

Yes I am assuming that! Shouldn't I?

No.

I'm supposing, as I said, that there is no slipping, so the friction force is a static friction force. The static friction force always equals (in magnitude) the force that wants to provoke movement, until it reaches its maximum possible value (which is the static friction coefficient times the component of the weight normal to the plane of movement).

OK.

So, if instead of a wheel we had a rectangular block (of equal mass as the wheel), the block would not be moving (assuming an initial state of rest, and assuming, of course, that there's no toppling), because the friction force would be equal to (in magnitude but opposing) the component of gravity parallel to the plane.

OK. But that's true for a block but not a wheel.

In both cases (toppling block or rolling wheel), there is no slipping, doesn't this mean (and is necessary for there to be no slipping) that the friction force has equal magnitude to the gravity component parallel to the plane?

No.

In the case of the wheel, you must figure out (using Newton's laws) the friction force necessary to have the wheel not slip. It's not simply equal to the gravity component parallel to the plane.

Set up the torque/force equations for rotation and translation and solve for the needed friction force.

 

[BobbyBear]

No.
In the case of the wheel, you must figure out (using Newton's laws) the friction force necessary to have the wheel not slip. It's not simply equal to the gravity component parallel to the plane.

Set up the torque/force equations for rotation and translation and solve for the needed friction force.

Well this comes to me as a bit of a surprise, as I'm thinking of the case of the wheel as a limiting case of the block of n sides (regular polygon) as the No. of sides goes to infinity.

So okay, let me accept that for the case of the wheel the friction force is not equal to (in magnitude) the gravity component parallel to the plane. So what you're saying is I'd apply Newton's laws of rotation and translation to the wheel, so I'd get on the one hand the angular acceleration of the wheel as a function of the unknown friction force f, and on the other the linear acceleration of the centre of mass (as a function of f again), and then impose the no-slipping condition through which I can relate the angular acceleration of the wheel to the linear acceleration of the centre of mass (as I did previously except that I had assumed f to be known), and from there solve for f. I think this would be the procedure you're indicating?

The fact that f is not equal to the component of gravity parallel to the plane would certainly eradicate the contradiction I was facing about the linear acceleration of the wheel being nonzero while the net linear force was (as I was erroneously assuming to be) zero.

But I'm still a little confused then about the case of the rectangular block (or n-sided polygon in general). You agreed that in this case, the friction force does equal the gravity component parallel to the plane if we are assuming there is no slipping. So wouldn't I end up with the same contradiction as I was earlier with the wheel? The net linear force on the block is zero, but IF it is toppling because there is a net moment, then the centre of mass would be accelerating . . . yes?

 

[Doc Al]

Well this comes to me as a bit of a surprise, as I'm thinking of the case of the wheel as a limiting case of the block of n sides (regular polygon) as the No. of sides goes to infinity.

That's an interesting way of looking at it, which I haven't analyzed yet.

So okay, let me accept that for the case of the wheel the friction force is not equal to (in magnitude) the gravity component parallel to the plane. So what you're saying is I'd apply Newton's laws of rotation and translation to the wheel, so I'd get on the one hand the angular acceleration of the wheel as a function of the unknown friction force f, and on the other the linear acceleration of the centre of mass (as a function of f again), and then impose the no-slipping condition through which I can relate the angular acceleration of the wheel to the linear acceleration of the centre of mass (as I did previously except that I had assumed f to be known), and from there solve for f. I think this would be the procedure you're indicating?

Exactly. This is how one can solve for the acceleration as well as the friction force.

The fact that f is not equal to the component of gravity parallel to the plane would certainly eradicate the contradiction I was facing about the linear acceleration of the wheel being nonzero while the net linear force was (as I was erroneously assuming to be) zero.

Right.

But I'm still a little confused then about the case of the rectangular block (or n-sided polygon in general). You agreed that in this case, the friction force does equal the gravity component parallel to the plane if we are assuming there is no slipping.

I agreed that the net force is zero in the case of a block that neither slipped nor toppled.

So wouldn't I end up with the same contradiction as I was earlier with the wheel? The net linear force on the block is zero, but IF it is toppling because there is a net moment, then the centre of mass would be accelerating . . . yes?

If the block is toppling, the net force is not zero.

 

[BobbyBear]

If the block is toppling, the net force is not zero.

So, even if the static friction force is capable of opposing the component of gravity parallel to the surface (in the sense that the maximum static friction force is greater than that force), it doesn't oppose it fully because of the geometry and weight being such that toppling occurs. Hmm this is very curious indeed, and though it must be so to satisfy Newton's laws, I'm wondering if there is an intuitive and logical physical explanation for this phenomenon.

Could it be that perhaps during toppling, what's happening is that the friction force no longer obeys the law f_{max}=\mu N, and that the friction force is the maximum it can possibly be? I do realize that the law f_{max}=\mu N is more of an approximate quantification of a complex phenomenon that may require a study of the principles of friction to understand better.

 

[Urmi Roy]

Well, I hope this doesn't sound too naive,but I remember that I once did a problem in which a block placed on a tilted ramp toppled over,and it seems that in that problem,it was considered that the block was put on a ramp,which was at a particular angle,such that considering the lowermost point of the block which is in contact with the ramp to contain the axis (going through that lowermost edge of the block), the net torque about that point due to the friction and the overall gravitational force was not zero.

This was mainly due to the angle at which the ramp was tilted--above a certain angle, the torques due to the friction and the overall gravitational forces are in such an orientation,that their torques no longer sum up to zero.

 

[Urmi Roy]

I'm thinking of the case of the wheel as a limiting case of the block of n sides (regular polygon) as the No. of sides goes to infinity.

Well,this aspect never struck my mind,so I never really thought in this direction--it opens a new route for thought.

So okay, let me accept that for the case of the wheel the friction force is not equal to (in magnitude) ...and then impose the no-slipping condition through which I can relate the angular acceleration of the wheel to the linear acceleration of the centre of mass (as I did previously except that I had assumed f to be known), and from there solve for f. I think this would be the procedure you're indicating?

I agree this is the best way to deal with it mathematically,but again, we are assuming that there is no slipping and then proceeding on these lines--I would rather like to refer to this in a less mathematical way,and instead,try to find out what is actually going on with all our forces,and their individual effects.

In our book, we find an entire section, called the "Paradox of the rotating wheel" in which the big question is, why does the wheel not accelerate inspite of the static frictional force providing the requisite torque. Here, the author brings in rolling friction,which according to him provides an opposing torque and hence the wheel does not accelerate, but as stated earlier, since we are considering rigid bodies, we ideally should not be bringing in rolling friction,and as clarified by BobbyBear, rolling friction is not an independent force.Further, in all other sources,they don't state anything about rolling friction.

Now, Doc Al said that the wheel on the ramp does indeed accelerate--perhaps so,but a similar situation arises in case of a wheel on level ground,where the force provoking the rolling of the wheel and the frictional forces both provide accelerating torque--but still the wheel does not accelerate.

This was actually the issue addressed in my book,that I referred to earlier.

Again I admit that frictional forces doing work sounds pretty bizarre,and as Doc Al clarifed,it doesn't actually do any work,however I fail to understand how this is so,since the friction does try to provide the wheel angular acceleration,even in the case of a wheel on level ground.According to Newton's second law applied to rotational mechanics,the friction should be doing work in this case.

 

[vin300]

A force acting at any point in the block has a conjugate force(opposite force) acting at a point exactly at the same distance from centre of mass, on the line joining the centre of mass to the point, but the other side of the CM.
That provides a torque, but the net force is zero.
Now the picture is such that the center of mass changes position because the gravitational force has a component along the length of the inclination.

 

[vin300]

Could it be that perhaps during toppling, what's happening is that the friction force no longer obeys the law f_{max}=\mu N

Yes, it does not obey the law of static friction.Kinetic and rolling are both coming into picture.
It is lesser than the static friction and the result is even lesser than something called rolling or kinetic friction as in the steady state, all of its underside is in frictional contact with the surface and when it's toppling, it comes in contact with the surface only after intervals.

 

[vin300]

and that the friction force is the maximum it can possibly be?

That's true

I do realize that the law f_{max}=\mu N is more of an approximate quantification of a complex phenomenon that may require a study of the principles of friction to understand better.

Not so easily. Let us have some more discussion.

 

[vin300]

rolling friction is not an independent force.

what do you mean by rolling friction is not an independent force?
it is similar to the other two frictional forces what differs is the state of motion.

 

[Doc Al]

In our book, we find an entire section, called the "Paradox of the rotating wheel" in which the big question is, why does the wheel not accelerate inspite of the static frictional force providing the requisite torque. Here, the author brings in rolling friction,which according to him provides an opposing torque and hence the wheel does not accelerate, but as stated earlier, since we are considering rigid bodies, we ideally should not be bringing in rolling friction,and as clarified by BobbyBear, rolling friction is not an independent force.Further, in all other sources,they don't state anything about rolling friction.

Please describe this "Paradox of the rotating wheel". What book are you using?

Now, Doc Al said that the wheel on the ramp does indeed accelerate--perhaps so,but a similar situation arises in case of a wheel on level ground,where the force provoking the rolling of the wheel and the frictional forces both provide accelerating torque--but still the wheel does not accelerate.

This was actually the issue addressed in my book,that I referred to earlier.

Please describe this situation in detail.

Again I admit that frictional forces doing work sounds pretty bizarre,and as Doc Al clarifed,it doesn't actually do any work,however I fail to understand how this is so,since the friction does try to provide the wheel angular acceleration,even in the case of a wheel on level ground.According to Newton's second law applied to rotational mechanics,the friction should be doing work in this case.

Static friction acting on the wheel does no work--the displacement of the point of contact is zero. However, Newton's laws still apply and that force does contribute to the wheel's acceleration. But don't confuse that with doing work--the ground is not an energy source.

 

[vin300]

Isn't the work done by a body to produce a displacement greater on rough sufaces than smoother sufaces because the forces of friction are different?
Yes, you do more work,but that doesn't go into the motion of the body. It appears on the underside of the body.
And why are frictional forces basically said to be electromagnetic?
When a linearly moving body comes in contact with a frictional surface, what was previosly only linear motion now becomes rotational too, the total momentum remaining the same.maybe this could explain why no work is done by the force.

 

[BobbyBear]

Well, I hope this doesn't sound too naive,but I remember that I once did a problem in which a block placed on a tilted ramp toppled over,and it seems that in that problem,it was considered that the block was put on a ramp,which was at a particular angle,such that considering the lowermost point of the block which is in contact with the ramp to contain the axis (going through that lowermost edge of the block), the net torque about that point due to the friction and the overall gravitational force was not zero.

This was mainly due to the angle at which the ramp was tilted--above a certain angle, the torques due to the friction and the overall gravitational forces are in such an orientation,that their torques no longer sum up to zero.

-nods- it is due to the angle of the ramp because the larger the incline, the bigger is the weight component parallel to the plane of movement. If the angle is suffiently small, the parallel weight component will be small and together with the static friction force (I am assuming no slipping) will only create a small moment, which if counteracted by the moment the normal-to-the-plane forces are capable of providing, will mean that the block won't topple (and thus the friction force will equal, in magnitude, the weight component parallel to the plane). Go on increasing the incline: the weight component parallel to the plane will increase, so will the moment it and the frictional force produces, whereas the weight component normal to the plane and the normal reaction force of the plane decrease, so decreasing the maximum resisting moment that this pair of forces can provide - until finally the block will topple.
I think that's right, no?

I still find it a little odd to come to terms with the phenomenon that just under the critical angle at which toppling occurs, the static friction force is equal to the weight component parallel to the plane, whereas for an angle just above the critical angle, the static friction force must take on a smaller value. DocAl has explaned why this must be so mathematically, I'm just wondering if there is an intuitive physical explanation of why this happens.

I also wanted to confirm one tiny thing: the resisiting moment produced by the weight component normal to the plane and the normal reaction of the plane produce a moment because the latter force is displaced from the perpendicular line to the plane that passes through the centre of mass of the block, right? The weight component though must always be applied at the centre of mass - it's the reaction force that goes on distancing itslef toward the edge of the contact region to create the moment - until it reaches the edge of the plane of contact between the block and the plane and there can be no further increase of the moment this pair produces. Thus the maximum resistive moment they can produce is the product of the normal force times the distance in the direction of the plane between the centre of mass and the contact edge.

 

[BobbyBear]

A force acting at any point in the block has a conjugate force(opposite force) acting at a point exactly at the same distance from centre of mass, on the line joining the centre of mass to the point, but the other side of the CM.

Yes okay maybe:P But from the perspective of rigid body mechanics, which is what we're discussing here, we're not considering how the forces are distributed internally within the solid - we're only concerned with their resultants, no? The resultant and the line of action is all that we need consider..

 

[BobbyBear]

Yes, it does not obey the law of static friction.Kinetic and rolling are both coming into picture.
It is lesser than the static friction and the result is even lesser than something called rolling or kinetic friction as in the steady state, all of its underside is in frictional contact with the surface and when it's toppling, it comes in contact with the surface only after intervals.

Isn't Kinetic friction when there is slipping (relative movement between the contact points)? If there's no slipping, then isn't it still static friction? It wouldn't obey the law f_{max}=\mu N, but wouldn't it still be static friction? And that's just what I mean, <br /> f_{max}=\mu N<br />
is just one attemp at quantifying friction, that was obtained experimentally, and usually works well and gives us a good idea of how large the friction force can be - but it depends on so many factors: on the temperature, on the type of surfaces, the time the bodies have been in contact, the relative speed (if the case of kinetic friction) . . . and for some materials/situations I recall seeing laws such as
<br /> f=\mu N^n<br />
where n is an exponent different to 1.
I don't know much about this or about the microspocic mechanisms that explain friction. But these are not universal laws like the laws of motion are, they are more like constitutive laws that we need in order to complete our sets of equations. That's the loose idea I was trying to portray . . .
And the non-universality of these laws is by which I was trying to justify that for rolling (or tumbling) witohut slipping, the static friction does not obey the law
<br /> f_{max}=\mu N<br />
in the sense that friction doesn't attain its maximum capable value according to that law.

 

[BobbyBear]

what do you mean by rolling friction is not an independent force?
it is similar to the other two frictional forces what differs is the state of motion.

I mean that friction in general is, independently of the different microscopic phenomena taking place, simply the force that opposes the relative movement between bodies at the contact points. Thus I don't understand 'rolling friction' as anything other than a combination of friction forces and non-friction forces (basically the forces normal to the contact surface at each point, which provide resistive moments) that occur when a particular object is rolling over a surface. (It's because it doesn't involve just friction forces that I'd rather simply call it rolling resistance). And because these forces arise due to the deformability of the objects in contact, they depend upon such things as the materials these objects are made of, etc.

I'd like assurance though that for perfectly rigid bodies, the concept of 'rolling friction' is meaningless . . . it would be zero always - which is corroborated by the fact that a rigid body that is rolling without slipping is not dissipating mechanical energy as heat due to friction. If you talk of 'rolling friction', you'd say that such a body is losing mechanical energy due to friction.

 

[BobbyBear]

In our book, we find an entire section, called the "Paradox of the rotating wheel" in which the big question is, why does the wheel not accelerate inspite of the static frictional force providing the requisite torque. Here, the author brings in rolling friction,which according to him provides an opposing torque and hence the wheel does not accelerate, but as stated earlier, since we are considering rigid bodies, we ideally should not be bringing in rolling friction,and as clarified by BobbyBear, rolling friction is not an independent force.Further, in all other sources,they don't state anything about rolling friction.

I'm inclined to agree with this: it's not really a paradox, simply that if bodies were really infinitely rigid, then a rolling wheel would keep on accelerating because there is a net moment upon the wheel (no opposing moment to counteract the one that set the wheel in rotational motion). It doesn't, of course, because the contact area deforms creating an opposing moment (rolling resistance). If you try and roll a deflated tyre, you will encounter a greater opposition than if you try and roll an inflated one. It's maybe in principle inconsistent to consider rolling resistance when studying rigid body mechanics because we can't explain how the rolling resistance came about, and we have to assume a certain value of this resistance - but since one thing are the forces/moments we consider when applying Newton's laws, and another how they originated, we can do this. But, you have to assume or accept a certain value of the 'rolling resistance' force, or obtain it through a simplified equation such as the one for the normal friction force but with a different (lower) friction coefficient.

We'd be assuming a rolling resistance effect so that our problem is closer to reality, while still assuming bodies to be rigid for the purpose of deducing their movement, which allows us to use concentrated forces -- maybe therein lies the paradox.

As DocAl said, it would be nice if you could describe what the book says.

 

[Urmi Roy]

Okay, here is an extract from the book I'm referring to...

It starts off with the big question: Why does a rolling sphere slow down.

Then,coming to the main issue,it says...

"When a sphere is rolled on a horizontal table,it slows down and eventually stops.The forces acting on the sphere are a. weight mg, b. friction at the contact and c. the normal force by the table on the sphere.
As the centre of the sphere decelerates,the friction should be opposite to its velocity,that is towards left (there is a simple diagram of a wheel rolling towards the reader's right). But this friction will have a clockwise torque that should increase the angular velocity of the sphere.
There must be an anticlockwise torque that causes the decrease in angular veocity.

Infact when, the sphere rolls on the table, both the sphere and the surface deform near the contact. The contact is not a single point as we normally assume, rather there is an area of contact.The front part pushes the table a bit more strongly than the back part. As a result, the normal force (by the table on the sphere) does not pass through the centre of the sphere, it is shifted towards the right of the centre of mass.
This force,then, has an anticlockwise torque. The net torque causes an angular deceleration."

The book is 'Concepts of Physics' part 1, by H.C Verma (proffessor of IIT Kanpur).

Now, in reference to this, it seems that if we neglect rolling friction(which we should be doing in the study of rigid bodies), the sphere will accelerate,due to the torque of static friction even on level round--this is strange, as friction is supposed to be a dissipative force--it is not supposed to favour relative motion between the sphere and table.

 

[Urmi Roy]

Now the picture is such that the center of mass changes position because the gravitational force has a component along the length of the inclination.

I'm just having a little problem in understanding how the centre of mass changes position,since I read that the position of the centre of mass depends only on the structure of the body in concern and the individual positions of its infinitely small constituent masses--all of which remain unchanged in this case.

 

[Urmi Roy]

Yes, it does not obey the law of static friction.Kinetic and rolling are both coming into picture.
It is lesser than the static friction and the result is even lesser than something called rolling or kinetic friction as in the steady state, all of its underside is in frictional contact with the surface and when it's toppling, it comes in contact with the surface only after intervals.

Basically, for toppling, we are not talking very specifically about what kind of friction is involved--it might be anything,right?

 

[Urmi Roy]

Static friction acting on the wheel does no work--the displacement of the point of contact is zero. However, Newton's laws still apply and that force does contribute to the wheel's acceleration. But don't confuse that with doing work--the ground is not an energy source.

I'm sorry, I still don't completely get it...please elaborate a little further.

 

[Urmi Roy]

Isn't the work done by a body to produce a displacement greater on rough sufaces than smoother sufaces because the forces of friction are different?
Yes, you do more work,but that doesn't go into the motion of the body. It appears on the underside of the body.

This is applicable in the case of,say for example a box being dragged across the floor, where friction plays a plain role of opposing relative motion between the box and the floor--here, it's converting mechanical energy into heat energy and hence dissipating it. In this case, friction does its expected role of preventing motion.

However, in case of rolling,it is seen that the friction is responsible for acceleration of the wheel,so it's not dissipating energy in this case,instead, it's speeding the wheel up,which kind of appears as if its 'providing' energy to the wheel.

This is where I'm having the problem.

And why are frictional forces basically said to be electromagnetic?
When a linearly moving body comes in contact with a frictional surface, what was previosly only linear motion now becomes rotational too, the total momentum remaining the same.maybe this could explain why no work is done by the force.

Well, I'm a little confused if we can consider the initial linear momentum being conserved by having the final sum of angular and linear momentum equal to it..can we sum angular and linear momenta?

 

[vin300]

I'm just having a little problem in understanding how the centre of mass changes position,since I read that the position of the centre of mass depends only on the structure of the body in concern and the individual positions of its infinitely small constituent masses--all of which remain unchanged in this case.

I didn't mean centre of mass changes position within the body, it changes position in the observer's frame.
Whatever internal changes take place in the body, its CM does not change position in the observer's frame but changes in its own frame, so the observer interprets "no linear motion"

 

[vin300]

As the centre of the sphere decelerates,the friction should be opposite to its velocity

Exactly, the friction must be opposite to the velocity, that is the velocity of the body at the point of contact, and not the velocity of its CM.
Whatever the motion, friction is always "resistive".

that is towards left (there is a simple diagram of a wheel rolling towards the reader's right).

It is towards the right.

But this friction will have a clockwise torque that should increase the angular velocity of the sphere.

Isn't that stupidity by a proffessor in a technical institute?

The contact is not a single point as we normally assume, rather there is an area of contact.

It forms a curve of contact.

 

[BobbyBear]

Okay, here is an extract from the book I'm referring to..."When a sphere is rolled on a horizontal table,it slows down and eventually stops.The forces acting on the sphere are a. weight mg, b. friction at the contact and c. the normal force by the table on the sphere.
As the centre of the sphere decelerates,the friction should be opposite to its velocity,that is towards left (there is a simple diagram of a wheel rolling towards the reader's right). But this friction will have a clockwise torque that should increase the angular velocity of the sphere.

The book is 'Concepts of Physics' part 1, by H.C Verma (proffessor of IIT Kanpur).

Urmi, I think the inconsistency lies in the assumption that the friction force is to the left. As DocAl has explained in this thread, you deduce the friction force that is compatible with Newton's laws. If there is no external force or torque upon the wheel prompting its movement, and the wheel is rolling and not slipping, then the friction force must be zero. This is the only compatible solution to applying both Newton's law of linear and rotational motion to the wheel in which you must consider the friction force as an unknown to be solved for (although under the assumption that static friction is capable of preventing slipping even if in the end the friction force is zero). The wheel will carry on its motion with uniform angular velocity indefinitely: it will neither accelerate nor deccelerate.

Although it seems a little strange to think that the friction force in this case is zero, remember that this is what the rigid-body model is predicting. And as the author of your book explained, it's not what exactly really happens, as we know by experience the wheel would tend to slow down if there was no external force (or torque) in favour of provoking its movement. But under the ideal assumptions this is what would happen, and what approximately happens when the two surfaces are almost indeformable.

 

[BobbyBear]

Well, I'm a little confused if we can consider the initial linear momentum being conserved by having the final sum of angular and linear momentum equal to it..can we sum angular and linear momenta?

You'd sum up the kinetic energies of translation and rotation.

 

[BobbyBear]

Exactly, the friction must be opposite to the velocity, that is the velocity of the body at the point of contact, and not the velocity of its CM.
Whatever the motion, friction is always "resistive".

friction is indeed in the direction opposing the relative velocity at the point of conctact, though if I may add, in the case of static friction, the relative velocity at the point of contact is nil, so the direction of the friction force is such that it opposes the impending relative velocity, if you get me (ie. the relative velocity that would be if the surfaces were frictionless).

 

[vin300]

I'd like assurance though that for perfectly rigid bodies, the concept of 'rolling friction' is meaningless . . . it would be zero always - which is corroborated by the fact that a rigid body that is rolling without slipping is not dissipating mechanical energy as heat due to friction.

Rigid bodies rolling without slipping might not dissipate mechanical energy as heat, the forces acting may weaken its intermolecular forces which break down at a later point of time. The energy is conserved.

 

[vin300]

friction is indeed in the direction opposing the relative velocity at the point of conctact, though if I may add, in the case of static friction, the relative velocity at the point of contact is nil, so the direction of the friction force is such that it opposes the impending relative velocity, if you get me (ie. the relative velocity that would be if the surfaces were frictionless).

When the body comes in contact with the surface in a direction different from the normal reaction or weight, it exerts a force on the suface and the surface exerts an equal force on the body but this force does not in any way assist the motion of the body, so it loses energy.

 

[BobbyBear]

Urminess,

adding to what I said in reference to the paradoxical scenario from the extract of your book about the friction force having to be zero, the fact that the friction force is zero doesn't mean that there could be a nonzero static friction force if so required (eg if another external force were to come into play). That is, the maximum static friction force is greater than zero. In this case, the friction force is zero, but the wheel is rolling without slipping, thus the linear velocity of the centre of mass is related to the angular velocity of the wheel.

But if the maximum friction force was zero, then even though we'd have exactly the same forces acting upon the wheel, and the wheel would be rotating with the same constant angular velocity (which would be whichever intial angular velocity you like), the wheel would be slipping without rolling (or rolling and slipping, depending on the initial movement - but in any case the rolling would not be due to the friction and the two movements would be uncorrelated), and the linear velocity of its centre of mass would be constant but arbitrary.

How would the system "know" whether there is a capacity for friction or not, if in both cases the friction force (by requirement of the conservation laws) is nil? Well you could think how the system would respond if you were to apply an inifinitesimal force to the wheel - in the first case, a nonzero friction force would appear, so that the movement provoked by this infinitesimal force would be a rolling without slipping movement, but not so in the second case, as that infinitesimal force would contribute to the system's linear movement but not affect its rotational movement. Since the case we've described is an ideal limiting case, in reality the system would never be faced with this quandary, as it would always be somewhat away from the ideal case, and it would know in which situation it stands.