Is Force a Conservative Vector Field?

[gspsaku]

1. The problem statemees and given/known data
A ball has a potential energy of 200J. Once the ball reached the ground, it's kinetic energy was 175J. Is this possible and does this defy the conservation of energy?

Homework Equations

KE = PE

The Attempt at a Solution

I am not sure. Being in high school, we have always been taught that KE = PE but if PE is 200, how can KE be 175? Where is the other 25J

 

[PhanthomJay]

1. The problem statemees and given/known data
A ball has a potential energy of 200J. Once the ball reached the ground, it's kinetic energy was 175J. Is this possible and does this defy the conservation of energy?

Homework Equations

KE = PE

The Attempt at a Solution

I am not sure. Being in high school, we have always been taught that KE = PE

you may have been taught that in certain special cases, the change in KE is equal to the change in PE

but if PE is 200, how can KE be 175? Where is the other 25J

Were you taught anything about work and work - energy concepts?

 

[haruspex]

we have always been taught that KE = PE

I assume you mean that their sum is constant, i.e. lost PE equals gained KE, and vice versa. That is only certain idealised conditions. If you were to drop a cannonball and a feather from the top of the tower at Pisa, which would reach the ground first, and why?

 

[gspsaku]

I assume you mean that their sum is constant, i.e. lost PE equals gained KE, and vice versa. That is only certain idealised conditions. If you were to drop a cannonball and a feather from the top of the tower at Pisa, which would reach the ground first, and why?

The cannonball would reach first since the feather will experience more air resistance, correct?

 

[gspsaku]

you may have been taught that in certain special cases, the change in KE is equal to the change in PE Were you taught anything about work and work - energy concepts?

We were taught that in a vacuum or ideal situations, the change in PE is equal to the change in KE.

It is just a yes/no and true false question. So I assume I would say Yes it is possible and False, it does not defy the conservation of energy.

 

[PhanthomJay]

We were taught that in a vacuum or ideal situations, the change in PE is equal to the change in KE.

It is just a yes/no and true false question. So I assume I would say Yes it is possible and False, it does not defy the conservation of energy.

Ok, but it would be good to give your reasons for your answers. What does air resistance have to do with the lost 25?

 

[gspsaku]

Ok, but it would be good to give your reasons for your answers. What does air resistance have to do with the lost 25?

Well, with air resistance, the object would not reach the speed/velocity it would in a vacuum thus have less change in KE which explains why the KE is less than the PE. So does that mean all we need to do, if we were to include air resistance, is add the air resistance to the PE side so it's 1/2mv^2 = mgh + air resistance?

Again, thanks for pushing me to think beyond the simple yes/no and true/false nature of answer.

 

[PhanthomJay]

Well, with air resistance, the object would not reach the speed/velocity it would in a vacuum thus have less change in KE which explains why the KE is less than the PE.

good.

So does that mean all we need to do, if we were to include air resistance, is add the air resistance to the PE side so it's 1/2mv^2 = mgh + air resistance?

you have the right idea , but that is not quite right. Air resistance is an opposing force ,like friction. If the change in the KE plus PE is non zero, then work must be done on the system by other forces, like friction or air resistance. Can you modify your equation ? .

 

[epenguin]

As this is a quiz, I am not sure what answer they want. It is impossible to answer correctly unless you are allowed to add some words of explanation.

So I would say – you seem to have pretty well got there anyway, potential energy physically almost does not mean anything. The change of potential energy between two situations is physically meaningful. In other words you have to specify the point where the potential energy is zero. Now usually there is an obvious natural point from which to refer. In this case that is the ground. They ought strictly to have told you, but since a suitable reference point is often obvious enough, often it is not specified. So I think you cannot answer without saying 'if the potential energy is with respect to ground level...',. P.E might have been measured from the bottom of a cliff and the ball hits the ground at the top. The potential energy at the top of the cliff might have been 25J with respect to the bottom, and then you would have had exact conservation of energy.

But assuming what I bolded above there is a discrepancy. So far as any information you are given, energy is has not been conserved. And then you have been reminded that some of the energy has gone away as heat with words like "friction" "viscosity" "dispersion". Did the question give you any information about how much the object or the air had heated? They did not. Did you ever feel after being taught how energy conservation was an absolute law, and then finding that all-around with bouncing balls and so on it doesn't hold, and then being told it'd gone away in heat which you don't see, that there was some cheating? I bet. I hope so anyway.

Energy is only conserved when an object moves under forces that depend only on its position with respect to other objects exerting those forces. This system of forces is said to be "a conservative field of forces". It is said to be conservative, because such a field conserves total energy! So that's just words so far. It gets to be science not just words is when it is proved mathematically that in a system with forces depending on position only, this quantity PE + ½ mv² (remember PE is Force X distance moved) remains constant during any motion. Realisation that there is this useful 'invariant of motion’ is credited to Emilie du Chatelet seen in the avatar here
https://www.physicsforums.com/threads/conte-riccati-and-jakob-hermann.865072/ and briefly discussed.

Mathematically proved - how? The essence of it you have probably already done or soon will, when you do acceleration of an object under constant force. You will find in what you do there, that the change in mgh equals minus the change in mv. The advanced and general proof is just a fancy version of that really. To pretty well guess that the principle holds generally, consider for instance a bunch of stars group interacting gravitationally with each other according to an inverse square law, so more complicated than your starting example. Think of anyone of them moving a short distance and time under the overall forces. Over this short distance and time the positions of the other stars won't have changed much, so it is almost like they haven't changed at all, the force on our chosen star has remained constant. Just like our initial elementary example. There will be a little change in the short time of both potential energy and ½ mv², but no change of their sum. This is a good enough example for now and some time to come to convince that with systems of position-only dependent forces this sum is a conserved quantity.

So we can get a bit more scientific by saying that in your example the discrepancy must be due to nonconservative forces. Four example the force of viscosity of in air, gas or liquid depends on the velocity of the object in its direction of motion, not its position. (it happens that this force increases with the velocity, Which is why a falling object reaches a constant velocity (known as 'terminal velocity’). It's when the speed has increased enough the viscous retarding force has become equal to the downpulling gravitational force.)

It was only a hundred years or so after du Chatelet that it was definitively clarified and established scientifically that at the microscopic level of the movements of atoms and molecules all physical forces are conservative so that what was at first a law with restricted applicability is instead a completely general physical principle. It just seems not to be because these movements of atoms and molecules are not appreciable at macroscopic level except as heat.

I might as well add here that this is all connected with reversibility. If your falling ball bounces without losing any energy (called elastic) it will bounce back to its previous height exactly having the same velocity but in the opposite direction that it had when it was there before, likewise our star system if we reverse all the velocities exactly is predicted to go back to a configuration identical to where it was at an earlier time. On the other hand your non conservatively bouncing ball is not reversible because you cannot practically get behind all of the molecules of air or in the ball and pushed them all back in the exact opposite direction and velocity to that in which they are moving.

I think if you just tell your questioners the things I have bolded and tell them that it is however established the total energy is really always conserved, but the macroscopically observed kinetic energy has been changed to into increased kinetic energy of atoms and molecules - heat (and for that matter often potential energy as well in rearrangement of these molecules, for you will find that a ball bounced too often may lose its bounce) it will satisfy them you have understood what they want you to. Maybe somebody else has to hand and can link to an article by Richard Feynman where he talks about elementary school textbooks in California where they make the students parrot that energy is conserved and everything that happens is all energy sloshing around between different forms. And he gets angry about it, and points out that it all doesn't mean a thing to just say that energy is conserved which is relatively difficult to understand truly.

 

[jbriggs444]

It gets to be science not just words is when it is proved mathematically that in a system with forces depending on position only, this quantity PE + ½ mv2 (remember PE is Force X distance moved) remains constant during any motion

That is a bit of an over-statement. A field with forces that depend on position only (i.e. any field whatsoever) need not be a conservative field.
A field with central forces (i.e. pointing toward or away from a defined center) whose magnitudes depend only on distance from the center is sure to be conservative [modulo some quibbles about being integrable]

 

[epenguin]

A field with central forces (i.e. pointing toward or away from a defined center) whose magnitudes depend only on distance from the center is sure to be conservative [modulo some quibbles about being integrable]

I didn't know that. Can you give any examples? Preferably that one can see intuitively that it is so.

 

[gneill]

I can think of a few ideas to ponder when thinking about KE and PE.

The details of the path the ball took were not disclosed. It's not clear that it was in free-fall or though what medium. So:

- Choice of zero reference for PE
- Ball was not falling but rolling, so KE would be split between rotational and linear KE
- Friction (air resistance, fluid viscosity, rolling/sliding friction,...)
- Buffered landing (say landing on a spring)
- Observer in a difference frame of reference than the ground

 

[jbriggs444]

I didn't know that. Can you give any examples? Preferably that one can see intuitively that it is so.

I fear that we are hijacking the thread. However...

Suppose that we have a clock face. At the center the force vector is zero. Above center (toward the 12) the force is rightward/clockwise with a magnitude in Newtons equal to the distance from the center in meters. Below center, conversely, the force is leftward/clockwise with a similar magnitude. To the right the force is downward/clockwise, to the left the force is upward/clockwise. More succinctly, the radial component is always zero and the tangential component is always non-zero and clockwise everywhere other than dead center..

If we put a pebble on the end of the minute hand, the work done by the field force on the pebble as it goes clockwise around once will be non-zero. That means that the force is not a conservative vector field.

If, by contrast, you require that the force vector be radial and a function of distance from the center alone then it is clear that the force around any circular path around the center must be zero (force is always at right angles to motion). On any radial path, one can simply integrate the central force times incremental radial distance to get work done. On any diagonal path, no matter how complicated, it is clear that the work done can be found based on the radial component alone and that over any closed path, the work done for the part of the path leading outward will cancel with the that done on the corresponding part of the path leading back inward.

https://en.wikipedia.org/wiki/Central_force