Conservation of energy in a system

[sspitz]

I'm pretty sure I understand everything my book says about force and mechanical energy for point particles. I'm slightly confused about how this applies to systems of particles.

For particles:
The line integral of force over a path is the change in KE. For conservative fields you can set a zero to create a PE function. Therefore, in conservative fields, mechanical energy is constant.

For systems:
All of the above applies to the center of mass of the system with respect to external forces.

In light of this, a lot of the problem I used to do just blindly by conservation of energy seem confusing.

For example (Kleppner 4.4), imagine a square. Remove every part of the square that is a distance R or less from the top right corner. Allow a small block of slide on the track formed by the missing 1/4 circle. Release the block from the top of the track. How fast is it going at the bottom? All surfaces are frictionless.

In the past, I set the zero of PE at the bottom of the track. Momentum is conserved. Then conserve mechanical energy. The initial PE of the block equals the final KE of the block + final KE of the square. Two equations, two unknowns.

This method equates the mechanical energy of the individual elements of the system at two points in time.
However, I no longer see why this is right.

(1) Why is mechanical energy necessarily conserved for anything in this system at all? For the block: the normal force between the block and square is nonconservative. For the system: The normal force between the floor and the square is external and nonconservative.

(2) If mechanical energy is conserved for a system, shouldn't it be the energy of the center of mass that matters, not the energy of the individual parts? Why do the parts in this system have to conserve?

I'm also having trouble writing down all my thoughts clearly all at once, so there may be more in reply to any answers. Thanks in advance.

 

[Simon Bridge]

(1) Why is mechanical energy necessarily conserved for anything in this system at all? For the block: the normal force between the block and square is nonconservative. For the system: The normal force between the floor and the square is external and nonconservative.

Forces don't have to be conservative.

The forces you are talking about arise from conservative fields.

(2) If mechanical energy is conserved for a system, shouldn't it be the energy of the center of mass that matters, not the energy of the individual parts? Why do the parts in this system have to conserve?

The law has to work regardless of how the parts are connected - how does each bit know if it is part of a whole or not, and how do they know where the center of mass is?

Anyway - if only the com has to obey these laws, then it may be possible to gain energy from nowhere by clever rearrangement of the bits about the com... making a perpetual motion machine. Some of the overbalanced wheels seem to be trying to do just that.

However - each bit will not be closed system by itself, so they will gain and lose energy in interactions with other bits.

 

[sspitz]

Forces don't have to be conservative.

The forces you are talking about arise from conservative fields.

I'm not sure I understand your meaning. You mean nonconservative forces exist? Could you explain why the contact forces are conservative?

The law has to work regardless of how the parts are connected - how does each bit know if it is part of a whole or not, and how do they know where the center of mass is?

But there are cases where the com conserves mechanical energy, but the parts added together do not. For example, a completely inelastic collision. My understanding is that this happens because the contact force between the colliding elements is nonconservative.

So it seems to me that given two elements in a conservative field that exert a nonconservative force on each other, the com will conserve energy, but the elements will not.

 

[Simon Bridge]

The contact forces arise in reaction to conservative force-fields.

An example of a non-conservative force would be friction: work against friction depends on the path taken. Here's another one.

Strictly speaking you only get nonconservative effects when you have missed out some of the energy conversions. Sometimes it makes for easier calculations - for instance, when a ball-bearing whacks into window-putty and sticks ... that would be inelastic right? So you'd expect that the com would conserve energy and momentum then? But what if the putty is also stuck to the table? In this case, the com also does not act like it conserves anything.

The energy of the ball goes into deforming the putty and manifests as internal energies (heat etc). The putty wants to move but the momentum gets channelled to the table instead, which is heavy and the floor is rough - maybe it moves a micron?

Similarly with friction, some energy goes into heat and sound for example.

At the QM level you can have changes which, although all the bits conserve energy, you get entropy. But at this level we have stopped talking in terms of forces completely.

 

[asteriatic]

I can assure you that your confusion is completely normal and understandable. The concept of conservation of energy can be tricky to grasp, especially when applied to systems of particles.

Firstly, let's clarify the definition of conservation of energy. It states that within a closed system, energy cannot be created or destroyed, it can only be transferred or converted from one form to another. In other words, the total amount of energy in a system remains constant.

Now, let's apply this to the example you mentioned from Kleppner 4.4. In this system, we have a block sliding down a track formed by a missing quarter circle. The block is in contact with the square, which is in turn in contact with the floor. This creates a system of particles.

To answer your first question, mechanical energy is conserved in this system because there are no external forces acting on the system. The normal forces between the block and square, and between the square and floor, are internal forces and do not affect the total mechanical energy of the system. This means that the initial mechanical energy of the system (at the top of the track) will be equal to the final mechanical energy (at the bottom of the track). This is why your previous method of setting the zero of PE at the bottom of the track and using conservation of energy worked.

Now, for your second question, you are correct that the energy of the center of mass is what ultimately matters in a system. However, the individual particles in the system must also conserve their own energy in order for the total energy of the system to remain constant. This is because the center of mass is a representation of the collective motion of the particles, and any changes in the individual energies of the particles will affect the overall motion of the system.

I hope this helps to clarify your understanding of conservation of energy in systems of particles. Keep in mind that this is a complex concept and it may take some time and practice to fully grasp it. Don't hesitate to ask more questions and seek further explanations if needed. As scientists, we are always learning and striving to deepen our understanding of the world around us.