Energy is conserved in pure rolling motion. Then why does the ball stops its motion after some time. I think it's not the case of air drag only. Does all work gets Transferred to surrounding in the form of heat?
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3Surface deformation and nothing is truly flat or circular. In any case, you're directly equating a real situation to a theoretical ideal one which is a problematic approach. – DKNguyen Jan 05 '22 at 14:29
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Possible duplicates: https://physics.stackexchange.com/q/661322/2451 and links therein. – Qmechanic Jan 05 '22 at 15:36
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Yes, energy is always conserved. If not then you have to widen your definition of the system to include all relevant parts.
For rolling motion, the main energy absorbers that cause slowing down is the contact with the surface and the mechanics of an axle (if its a wheel on a car e.g.). Kinetic energy might be lost as
- heat from friction in the axle bearings,
- work done to deform, compress and expand, a soft wheel material (think rubber),
- work done in deforming a soft surface (think sand beach),
and so on.
Also, due to the wheel and surface deforming slightly, the contact point will typically rather be a contact area over which the normal forces from all contact points might not necessarily point towards the wheel centre. Then such forces might introduce counteracting torques against the wheel motion, which slows it down.
Also, if the wheel "digs" a small pit in a soft surface, then it has to "carry itself" out of the pit constantly which means a fight against gravity with the posibility of even more forces, including kinetic frictions due to some sliding, being introduced. This slows the wheel down even more.
All such effects are often in one called rolling friction (which is slightly misleading since they are not necessarily frictions). And you have yourself mentioned air drag as yet another factor which becomes relevant at high speeds.
Steeven
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Energy is conserved when a real wheel or a real ball rolls along a real surface, but purely mechanical energy is not conserved. Some of the mechanical energy is converted to heat, which is considered to be another form of energy.
One important mechanism of energy conversion is hysteresis. No object is truly rigid. A rolling wheel or ball is continually deformed by its own weight (and in the case of a wheel on a road vehicle, by the weight of the vehicle and its cargo). The continually changing strain within the material converts mechanical energy into heat.
Solomon Slow
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If the energy were indeed conserved, the ball would roll without stopping. The reason why it stops is Rolling friction, which results in converting some of the mechanical energy into heat. Note that this is a type of friction that is different from more familiar static friction, which is also present in case of an object rolling, but which does not result in any dissipation.
Remarks:
- We are talking here about the mechanical energy of the macroscopic motion. Heat is also a form of energy, and to a large extent is also mechanical, but not macroscopic and therefore not readily visible.
Roger V.
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So, for practical cases energy is not conserved? like because of rigidity of material – MWD Jan 05 '22 at 15:10
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1Mechanical energy is not conserved. The overall energy (mechanical, plus heat, plus radiation plus whatever else) is always conserved. – Roger V. Jan 05 '22 at 15:14
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Rolling resistance is an interesting topic. It is clear from experience that a rolling wheel does eventually slow down and stop. So clearly there is some dissipative process that removes the energy from the wheel and transfers it to the environment. However, it is more than that, momentum and angular momentum are also conserved, and you have to consider mechanisms that dissipate them all.
For example, you mention air resistance. Air resistance doesn't produce a torque (or at least not obviously) so by itself it cannot be the sole cause. Also, we would expect that a tire rolling on the moon would stop too.
There is gravity, but on a level surface gravity produces neither torque nor work, and together with the normal force there is no change in momentum either.
There is also the friction force at the surface of the road. This acts backwards so it could account for the decrease in momentum, but naively the point of contact is not moving so the ordinary friction force does no work. Furthermore, the torque produced by the friction force points in the wrong direction and would actually tend to increase the angular momentum.
So the mechanism of rolling resistance is not trivial. The key is to recognize that the material of the tire deforms at the contact patch. Furthermore, this deformation and the forces involved in the deformation are asymmetric as shown in figures 2.8 and 2.11 here. Note how the forces on the leading edge of the contact patch are larger than the forces on the trailing edge.
This gives rise to a torque in the correct direction (reducing angular momentum), a net force in the correct direction (reducing linear momentum), and because the material at the contact patch deforms a negative mechanical power (reducing mechanical energy).
Dale
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Re, "friction...at the surface of the road...acts backwards." For an ideal wheel rolling on an ideal surface, there is no friction at all because there is no sliding contact between the wheel and the surface. For something like a car tire that deforms to meet the road over a relatively large "contact patch," the shape of the rubber will be continually changing while it is in contact with the pavement. Would that not cause friction forces that act in different directions at different points within the contact patch? – Solomon Slow Jan 05 '22 at 15:53
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@SolomonSlow yes, but it is clear that the net friction force is backwards from the fact that the acceleration is backwards. – Dale Jan 05 '22 at 16:14
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In an ideal case, once rolling starts, friction stops acting as there is no relative motion at the point of contact and the ball keeps on rolling. In a real life scenario, there is some deformation at the point of contact, and the normal shifts slightly and no longer passes through the centre. On performing torque analysis at the centre of mass, you will notice that the force of friction(acting opposite to the velocity of the centre of mass of the ball so as to slow it down), acts to increase the angular velocity of the ball, however, the slightly shifted normal force now provides a torque in a sense opposite to that of the angular velocity, therefore slowing the ball down eventually.
