Rotational Mechanics:Small Problems

[BobbyBear]

A rigid body that is rolling without slipping does dissipate mechanical energy as heat.

o:
This crushes all my notions:( Please explain? And, please tell me if you're considering there to be "rolling friction", maybe we're just thinking of different things? I'm assuming there is no "rolling friction", because the contact surfaces are not deforming, and contact occurs at a single point . . . although as I've discussed earlier, one could consider 'rolling friction' (for the sake of approximating reality better) as well as the bodies to be perfectly rigid (for the sake of applying Newton's laws using concentrated forces instead of having to consider internal stresses and what not), in which case I agree that there'd be energy dissipated as heat even though the wheel is not slipping, but due only to the 'rolling friction' which is really an attempt to quantify the energy that goes into deforming the surfaces.

 

[Doc Al]

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."

OK, this is just a description of rolling friction. The deformable surface is "bunched up" a bit ahead of the rolling sphere, which changes the direction of the force it exerts on the sphere.

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

I'm not familiar with that one.

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.

Neglecting rolling friction, there are no friction forces acting on the rolling sphere. It would just keep rolling.

 

[vin300]

o:
This crushes all my notions:( Please explain? And, please tell me if you're considering there to be "rolling friction", maybe we're just thinking of different things? I'm assuming there is no "rolling friction", because the contact surfaces are not deforming, and contact occurs at a single point . . . although as I've discussed earlier, one could consider 'rolling friction' (for the sake of approximating reality better) as well as the bodies to be perfectly rigid (for the sake of applying Newton's laws using concentrated forces instead of having to consider internal stresses and what not), in which case I agree that there'd be energy dissipated as heat even though the wheel is not slipping, but due only to the 'rolling friction' which is really an attempt to quantify the energy that goes into deforming the surfaces.

Before I could edit it understanding the mistake, it said I've to login again and the statement remained.
The contact surface must break apart after some time if the friction is too much as I said earlier.

 

[Doc Al]

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.

As long as we are just talking about static friction (and ignoring rolling friction due to deformation of the surface), there is no relative motion between the contact point and the surface. No slipping means no displacement and thus no work being done. Work is done by kinetic friction (and rolling friction), not by static friction. This is why you can apply conservation of mechanical energy to the wheel rolling down the incline--there are no dissipative forces (the friction is static).

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.

When the wheel rolls down the incline without slipping, static friction acts up the incline. That friction does two things:
(1) It slows down the translational motion of the wheel. (The net force on the wheel down the incline is less than it would be on a frictionless surface, thus the wheel's acceleration is less.)
(2) It increases the rotational speed of the wheel, since it applies a torque.

The friction doesn't provide energy, it just allows some of the gravitational energy to be converted to rotational kinetic energy.

Another example: When you walk or run (without slipping), again friction propels you forward yet it does no work and provides no energy. The energy is provided by your muscles; ground friction allows you to convert your internal chemical energy into translational kinetic energy. (If friction provided the energy, you wouldn't get tired. )

 

[BobbyBear]

​

As long as we are just talking about static friction (and ignoring rolling friction due to deformation of the surface), there is no relative motion between the contact point and the surface. No slipping means no displacement and thus no work being done. Work is done by kinetic friction (and rolling friction), not by static friction. This is why you can apply conservation of mechanical energy to the wheel rolling down the incline--there are no dissipative forces (the friction is static).

O: if the friction is static, how would there be kinetic friction as well? There either is relative motion or not..

 

[BobbyBear]

ooh, I think you just mean in general . . .
obviously the friction cannot be static and kinetic at the same time:P Sorry, I misinterpreted your meaning

 

[BobbyBear]

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.

Ya, I agree, friction cannot, by its very nature, increase the overall motion of an object, though ideally, if there is only static friction (and no deformation), there would be no dissipation of energy either. By definition static friction cannot do work! (another issue is whether in reality you'd have whatever other dissipating phenomena taking place).

 

[Doc Al]

O: if the friction is static, how would there be kinetic friction as well? There either is relative motion or not..

That's true. At any given time there's either static or kinetic friction, not both. I was just pointing out that work is done by kinetic friction, not static friction.

 

[vin300]

Ya, I agree, friction cannot, by its very nature, increase the overall motion of an object, though ideally, if there is only static friction (and no deformation), there would be no dissipation of energy either. By definition static friction cannot do work! (another issue is whether in reality you'd have whatever other dissipating phenomena taking place).

The post by me you're talking about above makes sense only if it is not static friction.

 

[vin300]

Are frictional forces said to be electromagnetic because they are associated with heat?

 

[Doc Al]

Are frictional forces said to be electromagnetic because they are associated with heat?

No. All contact forces, which are interactions between atoms and molecules, are fundamentally electromagnetic--as opposed to nuclear or gravitational.

 

[Urmi Roy]

It took a while to accumulate all these facts together in my head, but I think I've finally come to a conclusion. Please confirm if I'm right.

From what I figured, if we have a wheel, on level ground,on which a force is being applied tangentially,this force 'F' serves to accelerate the CM of the wheel as well as to supply a torque to the wheel.
The effect of this torque is felt on all the individual particles of the wheel,of which one is the lowermost point 'P', at which the wheel is in contact with the ground.

'F' tries to push this lowermost point to an adjacent position,but just as Doc Al pointed out, this point behaves quite like feet, running on the ground. In pushing against the ground due to the effect of 'F', the lowermost point 'P' experiences a reactional force from the ground due to the ground's friction, which resists its pushing past,and hence accelerating.
(A pair of feet running on the ground first push the ground and receive a reactional force due to friction from the ground).
However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),so it moves past its original position,but at uniform velocity(consiering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).

From the perspective of the CM, there is a net force 'F' so CM of the wheel accelerates.

Its the same thing for a wheel on a ramp,but here, 'F' is actually the component of gravitational force acting down the ramp.

All this is applicable only if the force with which 'P' tries to push past is less than the limiting friction.
If the force is greater than limiting friction,the wheel spins at a certain angular velocity depending on the effective torque and the linear velocity is determined separately by the net linear force ( by the way, can we have a force which only has an effective torque,but doesn't cause any linear acceleration of the wheel its working on--as in "an automobile with its engine revved to even 12000 rpm on a frictionless surface, which will stay put with an enormous angular velocity (measured at its wheels) but zero linear velocity."??).I suppose we can find out the angular and linear velocities imparted here separately.

Its true that it is difficult to imagine there to be no frictional force for a wheel rotating without slipping,and upon which there is no other force acting,but in this case,I suppose we can say that the 'P' doesn't have any tendency to 'push past' the ground, so in turn, the ground doesn't have to give any reactional force.

 

[BobbyBear]

From what I figured, if we have a wheel, on level ground,on which a force is being applied tangentially,this force 'F' serves to accelerate the CM of the wheel as well as to supply a torque to the wheel.
The effect of this torque is felt on all the individual particles of the wheel,of which one is the lowermost point 'P', at which the wheel is in contact with the ground.

'F' would not produce torque at the points of the wheel on its line of action.
For the solid to have a net torque, the torque at its centre of mass must be non-zero.
If 'F' is applied at the centre of mass of the wheel and there is no friction, then the wheel will not rotate. But if 'F' is applied somewhere other than the centre of mass, it will produce a net torque about the centre of mass and set the wheel rotating on its own.
When there is a friction force at 'P', it too produces torque at the centre of mass of the wheel. If 'F' is applied at the centre of mass, the it is the friction force at 'P' that is responsible for setting the wheel in rotational motion.
At least I think so.

However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),so it moves past its original position,but at uniform velocity(consiering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).

From the perspective of the CM, there is a net force 'F' so CM of the wheel accelerates.

I'm not following you:( How is there no net force on 'P'? If that were so then P would move in a straight line.
I don't understand what you're saying about the movements from different perspectives. Please explain?

by the way, can we have a force which only has an effective torque,but doesn't cause any linear acceleration of the wheel its working on--as in "an automobile with its engine revved to even 12000 rpm on a frictionless surface, which will stay put with an enormous angular velocity (measured at its wheels) but zero linear velocity."??).I suppose we can find out the angular and linear velocities imparted here separately.

Yes, but if there's friction it will tend to convert some of the angular movement into linear movement . . . the reverse of what it tends to do when the agent provoking movement is a force applied at the cenre of mass instead of a moment.

Its true that it is difficult to imagine there to be no frictional force for a wheel rotating without slipping,and upon which there is no other force acting,but in this case,I suppose we can say that the 'P' doesn't have any tendency to 'push past' the ground, so in turn, the ground doesn't have to give any reactional force.

I like that way of thinking about it :) 'P' doesn't have any tendency to push past the ground because the wheel has no tendency to accelerate - 'P' pushing against the ground would be to accelerate (or deccelerate) the wheel.

 

[Urmi Roy]

'F' would not produce torque at the points of the wheel on its line of action.
For the solid to have a net torque,...At least I think so.

I was just trying to say that the 'F' does effect the state of motion of 'P',and tries to accelerate it.

I'm not following you:( How is there no net force on 'P'? If that were so then P would move in a straight line.

Concentrating upon the instant that 'P' is the lowermost point,it does move in a straight line,doesn't it?

I don't understand what you're saying about the movements from different perspectives. Please explain?

Well, actually, I found that in rotational mechanics,one tends to say 'torque with respect to CM' or 'torque with respect to any other point',...quite like different frames of references, really.I don't know how appropiate that is in this case,but I just tried it out.

Yes, but if there's friction it will tend to convert some of the angular movement into linear movement . . . the reverse of what it tends to do when the agent provoking movement is a force applied at the cenre of mass instead of a moment.

...meaning,basically that this is possible,but friction is not such an example...did I understand that right?

Please bear with me if I'm being a little stupid, but I've always had an ambiguity with this topic.

 

[sganesh88]

Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of \frac{Fm}{M+m} is below the max static friction force)

However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),

Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :)

so it moves past its original position,but at uniform velocity(considering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).

No need to consider view points here. Just assume you're in an inertial frame and are provided with all facilities to measure the velocities and accelerations of individual particles as well as the wheel as a whole. Do check out what an inertial frame means if you don't already know. A final year Automobile engineering friend of mine once asked me, while i was explaining some stuff-and simultaneously doubting whether I'm right-, if such a frame called inertial frame really existed or if I'm just bluffing.
Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned, because anything above zero isn't admissible to the Earth's surface. Its just like me. tooo lazy!. Watch the cycloid Urmi.

 

[BobbyBear]

Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of \frac{Fm}{M+m} is below the max static friction force)

Ooooh tricky! that's something I need to ponder over:P:P However, yes, okay . . . but maybe now we need to redefine what 'doing work' is. The friction force would be doing work upon the top block, but it'd merely be transmitting part of the total work (the ratio m/M) beind done by the force F. It'd be acting like an internal force (so long as it's static friction). So maybe what we mean when we say that static friction cannot do work, is that we mean that it does not dissipate energy, thinking that friction in general (kinetic friction) is a dissipative force.
Yes?:P

 

[BobbyBear]

A final year Automobile engineering friend of mine once asked me, while i was explaining some stuff-and simultaneously doubting whether I'm right-, if such a frame called inertial frame really existed or if I'm just bluffing.

As far as I've been given to understand, it's Newton's first law that claims the existence of inertial frames.

 

[sganesh88]

Yes?:P

I'm afraid, no. Why are you bothered about static friction doing work anyway? Btw i was also irked when i heard gravity does work.

 

[sganesh88]

As far as I've been given to understand, it's Newton's first law that claims the existence of inertial frames.

Yes. So anyway first law created it. I didn't.

 

[BobbyBear]

Net force on individual particles

About the net force on 'P' (I'm assuming we are considering the situation that Urmi Roy described: the wheel subject to a tangential force 'F' (applied at its centre of mass?), and a static friction force at the point of contact 'P', thus the wheel has both a linear acceration and an angular acceleration (both of them compatible with the no slipping condition).

However, the 'F' keeps on acting and the net force on 'P' is zero (since F and friction at 'P' are opposite),so it moves past its original position,but at uniform velocity(consiering this from the point of view of the point 'P',it doesn't have zero velocity like it would appear to an observer at rest with respect to the ground).

Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :)
[...]
Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned, because anything above zero isn't admissible to the Earth's surface. Its just like me. tooo lazy!. Watch the cycloid Urmi.

I'm not sure how you'd know what forces are acting on individual particles of the wheel. Are you working it out from the movement you know of the wheel, from which you know the movement of each particle of the wheel, as we're considering the solid to be rigid?
So! basically, the centre of mass has a linear acceleration (and no angular acceleration), so the net force upon it is a linear force to the right.
And all points, including point 'P', have the same linear acceleration as the centre of mass, plus an angular acceleration about the centre of mass (superposition of two movemets). Thus, they are all subject to the same force that the centre of mass is, plus a centripetal force directed toward the centre of mass equal to the 'mass of the particle' times the distance of the particle to the centre of mass and the square of the angular velocity of the wheel at each instant.

 

[BobbyBear]

I'm afraid, no. Why are you bothered about static friction doing work anyway? Btw i was also irked when i heard gravity does work.

Oh :(
but I'm correct in saying that in your example the friction force doesn't do 'it's own' work, just transmits part of the work done by the force 'F'. And its true that friction cannot provoke movement on its own, it cannot transform some other kind of energy into kinetic energy! that's all I'm trying to say. At least this is true?

 

[Doc Al]

Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface. By pushing the lower block of mass M with a force F, the upper block of mass m, also accelerates because the static friction is doing work on it. (Assuming the value of \frac{Fm}{M+m} is below the max static friction force)

Good point. Whether a force does work on a system depends on the reference frame used to analyze the system. But the key point about static friction is that it's a passive force. For the "real" source of the energy used to accelerate the top block one must look to what's doing the work on the lower block.

 

[Urmi Roy]

Static friction does work in certain situations when the surface itself accelerates with respect to the observer. Consider the example of two blocks placed one above the other on a frictionless surface.

Thanks,this really helped to clear my concepts further!

Net force in the horizontal direction is zero,yes. Not in the vertical direction. Since you're considering the particles of the body-wheel- you have to consider the so-far-internal entity, namely the centripetal force which becomes an external force now. The rigidity of Newton's laws sometimes frightens me. :)

Will this have a serious bearing on the point of view I have formed about this entire event of roling without slipping?

No need to consider view points here. Just assume you're in an inertial frame and are provided with all facilities to measure the velocities and accelerations of individual particles as well as the wheel as a whole.
Back to our good ol wheel. The point P(the contact point of wheel with the ground) does come to zero velocity each time before being uplifted by the centripetal force. It cannot have the "uniform velocity" as you mentioned.

Just very cautiously,let me ask if this idea of considering view points,or rather frames of reference is actually wrong,even if I don't need it here.

Also, please tell me how I could modify my understanding of 'rolling' by stating what is wrong and what is right about my post(post 72 of this thread).

 

[BobbyBear]

Well, actually, I found that in rotational mechanics,one tends to say 'torque with respect to CM' or 'torque with respect to any other point',...quite like different frames of references, really.I don't know how appropiate that is in this case,but I just tried it out.

The moment produced by a force is not uniform in space, it depends upon the point you consider. Only the moment produced by a pair of forces of equal magnitude and opposite direction whose lines of action do not coincide produce a uniform moment in all of space.
That's why we talk of the moment (or torque) with respect to a point.

Maybe you're mixing this idea with different referece frames? I really don't think there's any relationship, but if you find there is let me know!

 

[sganesh88]

I'm not sure how you'd know what forces are acting on individual particles of the wheel. Are you working it out from the movement you know of the wheel, from which you know the movement of each particle of the wheel, as we're considering the solid to be rigid?

Assuming pure rolling;taking the mass of the particle to be some \Deltam, we can compute the instantanoues force acting on it at a particular position w.r.t COM.

Thus, they are all subject to the same force that the centre of mass is, plus a centripetal force directed toward the centre of mass equal to the 'mass of the particle' times the distance of the particle to the centre of mass and the square of the angular velocity of the wheel at each instant.

Considering pure rolling, the contact point P should have a zero horizontal velocity component, whatever the tangential force. The only unbalanced force acting on it is the centripetal force.

but I'm correct in saying that in your example the friction force doesn't do 'it's own' work, just transmits part of the work done by the force 'F'. And its true that friction cannot provoke movement on its own, it cannot transform some other kind of energy into kinetic energy! that's all I'm trying to say.

Going by the definition of work done which is, the dot product of the force and displacement, static friction can claim that it can do work. But yes, it can't cause motion on its own; i.e., without another force entering the picture.

 

[sganesh88]

Will this have a serious bearing on the point of view I have formed about this entire event of roling without slipping?

Rolling can be analysed without bothering about what happens at the "particle" level by just considering the wheel as a rigid body and computing the external forces and torques(w.r.t COM preferably) on it. But understanding the complex motions of particles is also quite fun.

Just very cautiously,let me ask if this idea of considering view points,or rather frames of reference is actually wrong,even if I don't need it here. Also, please tell me how I could modify my understanding of 'rolling' by stating what is wrong and what is right about my post(post 72 of this thread).

Reference frames are important when you describe motion. And you've come a long way from post #1. Much faster than me, i confess. Just keep thinking about it. Refer some books and internet articles. Anyway is this just for understanding purpose or are you giving some lecture on it?

 

[Urmi Roy]

Rolling can be analysed without bothering about what happens at the "particle" level ... But understanding the complex motions of particles is also quite fun.

I thought that perhaps analysing in this way might help in my understanding,but if you think it's not, I'll try in a different way.

Reference frames are important when you describe motion. And you've come a long way from post #1. Much faster than me, i confess.

BobbyBear said that the usage of 'frames of reference' in rotational mechanics isn't quite correct.Please tell me where I'm going wrong.

Just keep thinking about it. Refer some books and internet articles. Anyway is this just for understanding purpose or are you giving some lecture on it?

Pleeease don't leave it at that! I don't want to just improve in this,I want to finally remove this doubt that has been bothering me for so long,even though its just for the sake of my satisfaction,I don't have any lectures to give! I can only do it if you all help me!

 

[sganesh88]

I thought that perhaps analysing in this way might help in my understanding,but if you think it's not, I'll try in a different way.

Yes. it does. But to understand the motion of the wheel as a whole, the former approach would suffice.

BobbyBear said that the usage of 'frames of reference' in rotational mechanics isn't quite correct.Please tell me where I'm going wrong.

BobbyBear said reference frames aren't analogous to reference points w.r.t which moments are measured. Quantities like displacement,force, velocity and the whole lot of physical quantities lose their meaning in the absence of a reference frame. My initial doubt has been confirmed. Read about reference frames. Physicsforums and wikipedia would help you in it, I'm sure.

 

[sganesh88]

Pleeease don't leave it at that! I don't want to just improve in this,I want to finally remove this doubt that has been bothering me for so long,even though its just for the sake of my satisfaction,I don't have any lectures to give! I can only do it if you all help me!

What is the doubt that you presently have?

 

[Urmi Roy]

Actually I don't have anything new. As I said,this issue of 'rolling friction' has been bothering me for quite a while,the main difficulty of which stemmed from a particular extract of a book I read.

So,after getting together all that Doc Al, BobbyBear, vin300 and ofcourse, you said, I just tried to summarise the new picture in my head, in post 72.

Since then,you pointed out that viewing it from the aspect of individual particles isn't necessary, so apart from that, I just wanted to confirm my idea (as presented in #72) was basically right.

If you think I have any major problems, please modify my post and tell me the true 'picture'.