Total Mech Energy and Conservative forces

[sspitz]

I've managed to really confuse myself on the conservation of energy in a system. I guess the basic question boils down to why is the total mechanical energy of a system a meaningful quantity.

I understand E for a point particle in a conservative field is constant. I understand E for CoM with only conservative external forces is constant. Why is the sum of E for each element of a system constant? What are the specific conditions under which this is true?

At this point, words are only confusing me more, so I would prefer symbolic explanations using the definitions of work, ke, and potential. Here are some specific examples that confuse me.

#1 Elastic Collision: block 1 moving at speed v hits block 2, which is stationary. Equal mass blocks. Force between them is conservative (e.g. spring).

The following is wrong, but why?: look at block 1 in isolation. It is only subject to a conservative force. Therefore, its energy must be constant, yet it starts with E=KE and ends with E=0.

#2 Block on free wedge: a block slides without friction down a wedge, which slides without friction on the floor.

Is the normal force between block and wedge conservative? If yes, why does the total energy of the block decrease from the top of the wedge to the bottom? What is the PE of the normal force? How is that energy stored?

If no, how can total energy be conserved, since there is a nonconservative force?

Can anyone formulate and prove a statement for when the sum of the energy of each element in a given system must be concerned?

Thanks in advance

 

[amal]

Well, Mechanical Energy of particle remains constant in an conservative field because potential energy is converted into kinetic energy there. So, KE increases just as much PE decreases or vice versa. And this statement holds only when no non-conservative force is doing work on system.
Now, your first example. Here, final E is not 0. Because then the final velocity has to be 0 and hence frictional force is required to act. So, eventually, Non-conservative force acts on system.
The normal reaction is conservative as it does not depend on any sort of path taken to move the particle and total energy never decreases when object slides down an incline. You have said so in saying that total energy is conserved. Its only the PE that decreases. And, anyway, the normal force does not ever do any work as it is always perpendicular to velocity.

 

[sspitz]

Thanks for replying amal. I was hoping we could look a little more specifically at the examples.

#1: I'm not sure I understand your analysis of the system. My understanding is that the correct solution is that block 1 enters with velocity v. AFter the collision, block 1 stops moving, and block 2 leaves with velocity v. No friction required, just the spring between them. In light of this solution, my question is: take block 1 as the system. It only encounters a conservative force, the spring. Block 1 starts with KE = 1/2mv^2 and ends with KE=0. PE = 0 at the beginning and end. How can block 1 lose mechanical energy if it only encounters a conservative force? (Obviously, I realize the energy went to block 2 KE. However, this seems to violate the idea that a system must conserve mech E if it only encounters a conservative force.)

#2: First, it's my understanding that the normal force does work in this problem. The velocity of the block is not tangent to the slope of the wedge because the wedge also moves.

Second, I'm not sure I understand how to prove the normal force is conservative. It's not a vector field. If the block takes a different path, there is not normal force.

Third, if the normal force is conservative I see two problems. First is the same problem as in #1. The block only encounters conservative forces. However it finishes with less mech energy than it started with (i.e. some of the PE goes to the wedge). Second, where is the PE of the normal conservative force stored? What does the field look like? What is the potential function for the normal force?

Basically, the root question is the same for everything. Why is it justified to say the sum of mech energy of each element in a system is conserved when there are only conservative forces?

Lastly, I'll just say I'm not trying to be argumentative. I realize everything I've written is incorrect. Just trying to understand.

 

[amal]

Well see, a conservative force by definition is the one in which energy of system is conserved. And a system must have at least two objects. So it is wrong to see the block isolated in both cases as they don't make a system. Total energy of system is conserved. it is like saying that the block 1 stops after collision so its momentum is not conserved. Momentum of system is conserved and force is conservative.

 

[asteriatic]

I can understand how these concepts can be confusing. The conservation of energy is a fundamental principle in physics and it is important to understand how it applies to different systems and situations.

First, let's define what we mean by total mechanical energy. It is the sum of the kinetic energy (KE) and potential energy (PE) of all the elements in a system. In other words, it is the energy associated with the motion and position of all the objects in the system.

Now, for a system to have a constant total mechanical energy, there are two conditions that must be met: the forces acting on the system must be conservative and there must be no external forces acting on the system. Let's break this down further.

A conservative force is one that does not depend on the path taken by an object, only on its initial and final positions. Examples of conservative forces include gravity and the spring force. These forces can be described by a potential energy function, which is the energy associated with the position of an object in the presence of these forces.

Now, let's look at your examples. In the first example, the elastic collision, the force between the blocks is conservative (spring force). However, there is an external force (the ground) that is acting on the system. This external force is not conservative, which means that it can do work on the system and change its total mechanical energy. Therefore, the total mechanical energy of the system is not conserved in this case.

In the second example, the block on the free wedge, the normal force between the block and the wedge is not a conservative force. This is because the normal force depends on the contact between the two surfaces, and the contact point is changing as the block slides down the wedge. Therefore, this force can do work on the system and change its total mechanical energy. Additionally, there is an external force (gravity) acting on the system, which also contributes to the change in total mechanical energy.

To answer your question about when the sum of the energy of each element in a given system must be conserved, it is when the two conditions mentioned earlier are met: the forces must be conservative and there must be no external forces acting on the system.

I hope this helps clarify the concept of conservation of energy in a system. It is a fundamental principle that helps us understand the behavior of objects and systems in the physical world. If you have any further questions, please don't hesitate to