Defining Energy: Types, Einstein's Formula, & More

[student85]

Ok, I've been having many question concerning energy lately. First of all...energy is the hability a body has to do work, am I right? If not, how would you define it correctly?

Then, how many types of energy are there...kinethic and potential...is that it? What is the energy in einstein's formula E=mc2... is it the TOTAL energy? That includes mechanical energy (KE and PE) and other types? What are the other types?


Any help would be very appreciated, I really want to learn about this topic, thanks.

 

[Andrew Mason]

Ok, I've been having many question concerning energy lately. First of all...energy is the hability a body has to do work, am I right? If not, how would you define it correctly?

Then, how many types of energy are there...kinethic and potential...is that it? What is the energy in einstein's formula E=mc2... is it the TOTAL energy? That includes mechanical energy (KE and PE) and other types? What are the other types?


Any help would be very appreciated, I really want to learn about this topic, thanks.

Your questions are good ones. It shows that you are thinking.

It is not immediately obvious why energy (= force x distance) should be a particularly useful quantity. It was not really until the 19th century that it became widely used in physics when Joule proved that the heat generated by mechanical motion was proportional to the distance a falling weight moved. The historical treatment of energy is an interesting topic and well worth reading about. See, for example, http://en.wikipedia.org/wiki/Energy"

Einstein showed that the emission and absorption of light necessarily transfers mass m where $m = E/c^2$ , E being the energy of the light.

Einstein concluded that the mass of a body is a measure of its energy content: that there was an enormous amount of energy locked up on matter: E = 9x10^16 x mass (mks units) Subsequent work proved he was right.

AM

 

[quasar987]

Cool post AM.

 

[actionintegral]

Energy per se is not an important quantity. What makes energy important is the notion of "potential energy". Total Energy is defined as the sum of "kinetic energy" ( a measurable quantity ) + "potential energy" (a mathematically defined quantity). Under certain circumstances, Energy is "conserved" which makes certain problems easier to solve. Springs and falling bodies are examples.

 

[vanesch]

Energy is a many-faced item. It started out as a kind of curiosity in Newtonian mechanics: In certain specific cases (namely those where the forces were conservative, or derivable from a potential function), one could write a function of the mechanical configuration ("potential energy") and a function of the velocities ("kinetic energy"), and it turned out that the sum of both was conserved through time.
In Newtonian mechanics as such, this didn't need to be: one can introduce forces (friction forces) such that this trick doesn't work. But in the case of conservative forces, it was nice to have this "energy" trick which allowed to solve problems (especially together with "momentum conservation", which followed from Newton's action=reaction principle).

People reformulated Newtonian mechanics for the specific case that all forces were conservative and out of that came an entire body of theoretical work, which was Lagrangian and Hamiltonian mechanics. It is a beautiful framework, and assumes from the start that "energy is conserved". (ok, nitpickers will tell me that there are extensions to Lagrangian mechanics which allow dissipation functions, true...)

Now, what turned out to be a "limiting hypothesis" in the beginning, namely that we had to limit ourselves to conservative forces, showed up to be of great interest. Indeed, in the Hamiltonian formulation of mechanics, it turns out that this quantity, energy, plays a fundamental role as generator of time translations. It's a bit difficult to explain this here, but if a system possesses a certain continuous symmetry, then there is a "generator of the symmetry", which indicates how the system transforms under said symmetry operation. In an abstract way, one can think of a "time translation" also as a symmetry operation on the system, and it turns out that the corresponding generator is nothing else but "energy".
In a similar way, space translations correspond to a generator which is nothing else but momentum.

It also turns out that all fundamental interactions we know about, are conservative. This gives a special importance to the Lagrangian and Hamiltonian formulations, and hence, energy.

Comes in (special) relativity. In special relativity, it turns out that the 4-th component one has to add to the 3-momentum (the ordinary Newtonian momentum which was already postulated to be conserved, through the action-reaction principle) to make a good 4-vector, is nothing else but, again, energy. Given that, under Lorentz transformations, the 4-th component (the "time" component) mixes with the 3-vector (the "space" component) of a 4-vector, momentum cannot be conserved if energy is not conserved. So if we took (since Newton's time) momentum as conserved, and we accept relativity, this automatically means that energy is conserved - which is also an explanation for the fact that all fundamental interactions are conservative... and which confirms the importance of the Lagrangian formulation of mechanics.

From a totally different side came thermodynamics. The first law of thermodynamics states that there is a conserved state function, and this is recognized, in statistical mechanics, as, again, the total energy of the system. We now also understand the "non-conservative" forces we originally considered in Newtonian mechanics: they correspond simply to the transformation of macroscopic mechanical energy into microscopic mechanical energy. We only saw the "energy dissipate" because we forgot to take into account the microscopic degrees of freedom ; we can restore this by incorporating "heat" in the balance.

We thus see that the concept of energy grew in importance over the course of physics history:
from a curiosity in Newtonian mechanics, it:
- became a pillar in the Lagrangian and Hamiltonian formulation
- became the generator of time translations
- became the 4-th component of momentum in relativity
- became the state function in the first law of thermodynamics

 

[quasar987]

Encore!

 

[student85]

Thanks a lot for your posts.

I've not learned relativity or any of Einstein's works so here's my question:
ActionIntegral said that the total energy was equal to PE + KE. But if you equal Einstein's equation E=mc2 to KE + PE you obviously don't get the same thing. What's wrong here?

 

[quasar987]

Yeah $E=\gamma m_0c^2$ only encompasses the mass energy and the kinetic energy. In other words, it says that a free particle (no potential energy) does not simply have energy E=K, but rather, it has energy $E=m_0c^2 + K$ where K is the relativistic kinetic energy, which is not equal to mv²/2 but rather, defined as the difference btw the rest mass $m_0c^2$ and the total energy $E=\gamma m_0c^2$ (much like heat energy is defined as the difference in total energy variation and work)

Einstein's E=mc² only introduced a new term (the rest mass energy) that we must take into account when trying to keep track of what we call energy.

 

[student85]

Thanks quasar. One more thing, why is PE not considered there? PE is like a fictitous energy I think. It is not energy the body posseses but rather the hability that body has to gain energy, am I right?

 

[quasar987]

Put your particle in a (conservative) force field and the energy becomes $E=m_0c^2+KE+PE$. Happy?

 

[xAxis]

I've also never understood what is energy. I understand a PE as a kinetic energy that body can have due to force acting on it.
I would also like to know if diferent orbits around some planet have diferent energies and how do we calculate the energy associated with particular orbit?

 

[quasar987]

Yes, different orbits have different energies. For instance, consider a planet in circular orbit around a sun. You know from your mechanics class that for a particle to achieve circular orbit, it must have a speed that is purely tangential and whose magnitude is related to the central force keeping it in orbit by

$$a=\frac{F}{m}=\frac{v^2}{r}$$

here the force is gravitational in nature, so it is of the form $F=GMm/r^2$ and solving our equation for v(r) yields

$$v=\sqrt{\frac{GM}{r}}$$

This tells you that if you know the radius of the orbit, then you know its speed, and hence its kinetic energy. In particular, given an object in a circular orbit, the energy is as great as the radius of the orbit is small.

If you take a mechanics class in university, you will solve entirely the equation of motion for bodies in a gravitational field and you will find what all the possible orbits are and their related energies.

For a peek of what awaits you see

https://www.physicsforums.com/showthread.php?t=128711

 

[billiards]

In the context of fluid mechanics a particle has kinetic and potential energy, AND pressure energy (note this is a form of elastic potential energy
). Also chemical energy I don't think has been mentioned. Heat energy= kinetic energy i suppose. Can't think of any other energies right now ...