Conservation of Momentum and Energy

[Walter]

Starting a new subject because I don’t want to detract from the current discussion around CoAM.

There seems to be some misconceptions about the conservation laws that I wanted to clear up.

First, let’s review the laws of conservation of energy and conservation of momentum. Both deal with isolated systems, or systems that do not have external forces acting on them. In the case of CoM in the absence of external forces the momentum of our system is conserved. In the case of CoE in the absence of external and internal forces the energy of our system is conserved.

The first misconception I often see is that if an external force is acting on a system then one or both of the conservation laws don’t apply and, therefore, can’t be used for analysis of the system. Think about that. There is no system in the known universe, other than the complete universe, where external forces such as gravity don’t exist. Friction is also present in any system with moving parts. Does that mean that two of the foundational laws of classical mechanics have no useful purpose? Perhaps if we look at a corollary for these laws. i.e. if external forces are acting on a system then the momentum and energy of the system are not constant. Given that the laws say that energy and momentum cannot be created or destroyed that must mean that, because of conservation, we can determine the forces acting on a system by the changes in their momentum. The laws can’t tell us the exact nature or source of the force(s) but they tell us the magnitude of the net force. Given that force and momentum are vector quantities we can use CoM to determine both the magnitude and direction of the net force acting on the system. Furthermore, if the changes in momentum or energy of a system are due to net external forces only then the change in momentum and change in energy of the system will have the same magnitude of force acting on them, otherwise we know that there are internal forces as well as external forces at play.

To be continued…

[Walter]

For now all of my discussions will assume that we are talking about a tethered cast with level line of constant linear density.

I think we have all seen the CoM/CoE analysis of an unrolling line that assumes that, since the mass of our fly leg is decreasing, then CoM and CoE tell us that the speed of the fly leg must be increasing (eventually to infinity). This analysis results in a comparison of the fly leg speed due to CoM vs CoE and we find that the speed increase due to CoE is less pronounced than what we would see due to CoM then we have to use CoE as the basis for any further analysis and that use of CoM is wrong.

Unfortunately, this analysis is flawed because the initial description and assumptions of the problem are flawed. We know that the mass of our system is changing. How does that happen without some sort of force acting on our system? We also know that the speed of our fly leg never reaches anywhere near the types of speeds that are predicted by either conservation law in this model.

To be continued…

[Walter]

If the previous application of the conservation laws to the unrolling fly line is flawed then what is the correct analysis? We know that the speed of the fly leg remains fairly constant as it unrolls. I have seen measured casts where there is a small change in speed in the fly leg in the final stages of unrolling but, unfortunately, the database is small and these types of small changes could be attributed to a number of external factors such as wind or gravity. For my model I’m going to start with the assumption that the fly leg speed is constant. Obviously if the speed remains constant and the mass is changing then both the momentum and energy of the fly leg are changing and we can calculate the force acting in the line based on the observed change in those quantities. When we do that we see that the change in both momentum and energy is constant over time. In addition, we can determine that the force generating the change in both momentum and energy is the same so it must be an external force. The effect from internal forces is zero. Unfortunately, the conservation laws do not tell us the source of this force. I’m sure many of you can hazard a guess but I’ll just leave it at that for now. Here is the graphical output of the changes in energy and momentum assuming a level line moving at constant speed and the estimated force that would generate the changes in momentum and energy. Note that there are only three lines visible in the chart because the force calculated from CoM is the same line calculated from CoE.

[Merlin]

Hi Walter

There is a need to consider conservative forces (e.g. gravity) and non conservative forces (e.g. friction) when talking about CoE or CoM.

Merlin

[Walter]

Merlin,

Thanks for the response. So far I’ve avoided discussing the specific forces acting on the system. My goal was to show that we can’t arbitrarily reject either CoE or CoM in favour of the other as useful tools to analyze what we see happening. There are times when one will be more suitable than the other, just like there are times when Lagrange is preferable over Newton, and sometimes one path will lead to an intractable solution. But, in the case, where you can solve a problem in multiple ways you should always get the same answer.

With respect to CoE one of the issues is that energy can be converted into different forms. KE can be converted to PE and PE can be positional or a totally different form such as elastic potential, chemical potential or just heat.