Energy conservation problem

[User_54]

I have two equations:

 

[latex]M_1\ddot{x}_1=F,\qquad (1)[/latex]

 

and

 

[latex]\ddot{x}_2+\omega^2\cdot x_2=\epsilon\cdot\ddot{x}_1.\qquad (2)[/latex]

 

The first equation says that particle [latex]M_1[/latex] can be affected with some force [latex]F[/latex] and the second one describes the oscillation amplitude variations if the first particle is affected by a force. Oscillations take some energy. How to complete the first equation to take into account this energy loss?

 

Thanks.

Edited February 28, 2013 by User_54

[Enthalpy]

That's nothing more than my spontaneous gut feeling, but the factor epsilon models a coupling force between directions 1 and 2. If you replace this action from 1 on 2 by an interaction between 1 and 2, you should solve simultaneously the conservation of momentum (which is violated presently) and of energy.

 

Once this is done , a new main direction, obtained by a rotation somewhere between 1 and 2, will allow you to write the pair of equations as on acceleration+oscillation, plus a trivial one without movement on the perpendicular direction.

 

In case 1 and 2 are not geometric directions, the geometric analogy is still a guide to the mathematical operation.

[User_54]

In case 1 and 2 are not geometric directions, the geometric analogy is still a guide to the mathematical operation.

No, they are just different scalar variables. I would edit my post to avoid confusion, but it is not possible anymore.

 

Momentum cannot be conserved because of an external force [latex]F(x_1,t)[/latex] acting on the particle [latex]M_1[/latex].

 

My problem is to "guess" or "derive" the interaction between 1 and 2.

Edited March 1, 2013 by User_54

[Enthalpy]

It stays that x1 acts on x2 but x2 still doesn't on x1. If it's a question of particles, then these two scalar variables should interact.

 

Conservation of momentum: I mean in the interaction of x1 and x2, independently of the external F. If the coupling shall conserve momentum, then it must act symmetrically on both x1 and x2.

[Bob_for_short]

... these two scalar variables should interact.

That's my problem - to make them inetract. I do not know how.

[elfmotat]

I'm not exactly sure what you're asking. Are you trying to find the work done on particle 1 by [math]F[/math], so as to determine the energy of the system as a function of time?

[User_54]

I would like to obtain a self-consistent desctiprion. Particle-1 does not currently "know" about oscillator. Should it konow, and if so, how to take the energy loss into account?

[swansont]

If particle 1 exerts a force on particle 2, then apply Newton's Third law.

[User_54]

If particle 1 exerts a force on particle 2, then apply Newton's Third law.

If I add [latex]-\epsilon \ddot{x}_1[/latex] to the right-hand side of Eq. (1), then I will decrease acceleration of particle-1. On the other hand, if the oscillator phase is such that it "helps" accelerate particle-1, then this fact is not contained in the modified Eq (1). So something different is needed.

[swansont]

If I add [latex]-\epsilon \ddot{x}_1[/latex] to the right-hand side of Eq. (1), then I will decrease acceleration of particle-1. On the other hand, if the oscillator phase is such that it "helps" accelerate particle-1, then this fact is not contained in the modified Eq (1). So something different is needed.

 

Eq 1 is just Newton's second law. No need to modify it. The force on it is from the perturbation from particle 2. Unless it's a matter of not having been given enough information.

[User_54]

Eq 1 is just Newton's second law. No need to modify it. The force on it is from the perturbation from particle 2. Unless it's a matter of not having been given enough information.

Apparently the total energy is not conserved (in a potential force F). Is it possible to modify these equation in such a way that tha total energy is conserved? If so, what are the modified equations?

[swansont]

If there is a dissipative force, then energy won't be conserved. If you can account for the work, then you can do an energy treatment of the problem. But F=ma itself does not imply energy is not conserved within the system.

[User_54]

If there is a dissipative force, then energy won't be conserved. If you can account for the work, then you can do an energy treatment of the problem. But F=ma itself does not imply energy is not conserved within the system.

Let us consider a potential wall. Upon hitting the wall the particle reflects with the same energy, but the oscillator acquires energy too. Where does the latter come from?

Edited March 3, 2013 by User_54

[swansont]

Let us consider a potential wall. Upon hitting the wall the particle reflects with the same energy, but the oscillator acquires energy too. Wheredoes the latter come from?

 

It's easy to set up word problems that violate physical law. The particle cannot reflect with equal energy unless work is done.

[User_54]

It's easy to set up word problems that violate physical law. The particle cannot reflect with equal energy unless work is done.

I described a situation followed form the Eqs. (1) and (2). Where am I wrong?

[swansont]

F=ma (eq 1) says nothing inherently about energy conservation. If the transfer is internal to the system energy will be conserved.

 

Eq 2 appears to describe how system 2 is coupled to system 1. Where is the energy loss?

[Bill Angel]

I have two equations: [latex]M_1\ddot{x}_1=F,\qquad (1)[/latex]

and [latex]\ddot{x}_2+\omega^2\cdot x_2=\epsilon\cdot\ddot{x}_1.\qquad (2)[/latex]

The first equation says that particle [latex]M_1[/latex] can be affected with some force [latex]F[/latex] and the second one describes the oscillation amplitude variations if the first particle is affected by a force. Oscillations take some energy. How to complete the first equation to take into account this energy loss? Thanks.

If the particle [latex]M_1[/latex] appears in the second equation, and it interacts with a particle [latex]M_2[/latex] via that equation, should not the masses of particles [latex]M_1[/latex] and [latex]M_2[/latex] also appear in the second equation?

[User_54]

If the particle [latex]M_1[/latex] appears in the second equation, and it interacts with a particle [latex]M_2[/latex] via that equation, should not the masses of particles [latex]M_1[/latex] and [latex]M_2[/latex] also appear in the second equation?

We can multiply Eq. (2) by M₂ to have it, if you like.

 

F=ma (eq 1) says nothing inherently about energy conservation. If the transfer is internal to the system energy will be conserved.

 

Eq 2 appears to describe how system 2 is coupled to system 1. Where is the energy loss?

If an external force is time dependent (a moving wall), then F(t) acts during a short period and transfers some energy to M₁. This energy does not depend on presence or absence of oscillator in this set (1), (2). On the orher hand, the oscillator also gains energy due to acceleration of particle-1. Does it look like energy is conserved?

Edited March 3, 2013 by User_54

[swansont]

If an external force is time dependent (a moving wall), then F(t) acts during a short period and transfers some energy to M₁. This energy does not depend on presence or absence of oscillator in this set (1), (2). On the orher hand, the oscillator also gains energy due to acceleration of particle-1. Does it look like energy is conserved?

 

If there is work being done by an external force, no. But there is nothing in your problem that says anything explicit about an external force.

[User_54]

If there is work being done by an external force, no. But there is nothing in your problem that says anything explicit about an external force.

An external force can be x₁-dependent and time dependent. I gave two examples: a standing wall and a moving particle, and a moving wall and an initially still particle. In both cases the external force accelerates particle-1 and changes its velocity.

Edited March 3, 2013 by User_54

[Bill Angel]

We can multiply Eq. (2) by M₂ to have it, if you like.

 

 

If an external force is time dependent (a moving wall), then F(t) acts during a short period and transfers some energy to M₁. This energy does not depend on presence or absence of oscillator in this set (1), (2). On the orher hand, the oscillator also gains energy due to acceleration of particle-1. Does it look like energy is conserved?

Does the situation you are examining resemble that of a driven oscillator?

See http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/oscdr.html

[swansont]

An external force can be x₁-dependent and time dependent. I gave two examples: a standing wall and a moving particle, and a moving wall and an initially still particle. In both cases the external force accelerates particle-1 and changes its velocity.

 

Yes, you gave two possibilities, but neither is given in the original statement of the problem.

[User_54]

Does the situation you are examining resemble that of a driven oscillator?

 

Yes, to a great extent: Eq. (2) is a driven oscillator equation. The damping force is simply negligible here, I guess.

 

Well, I was given two equations written above and without much explanation about the total system. I am free to modify the system equations and give it a "proper' explanation.

Edited March 4, 2013 by User_54

[User_54]

So, no fruitful ideas on fixing the system of equations?

[User_54]

This situation is similar to a situation in the classical electrodynamics where a charge acceleration creates electromagnetic waves. Maybe the electrodynamical example can be helpful here?