Thursday, October 4, 2012

Anonymous asked: What are the 9 types of energy

Hey there,

Why don’t you check out this earlier post. It should give you a brief intro to the types of energy, the law of conservation of energy, efficiency and Sankey diagrams!

Hopefully that will help you out and thank you for the question!

As always, feel free to submit any other inquiries to our ask.

Friday, October 28, 2011
The Virial Theorem



In the transition from classical to statistical mechanics, are there familiar quantities that remain constant? The Virial theorem defines a law for how the total kinetic energy of a system behaves under the right conditions, and is equally valid for a one particle system or a mole of particles.



Rudolf Clausius, the man responsible for the first mathematical treatment of entropy and for one of the classic statements of the second law of thermodynamics, defined a quantity G (now called the Virial of Clausius):



G ≡ Σi(pi · ri)



Where the sum is taken over all the particles in a system. You may want to satisfy yourself (it’s a short derivation) that taking the time derivative gives:



dG/dt = 2T + Σi(Fi · ri)



Where T is the total kinetic energy of the system (Σ  ½mv2) and dp/dt = F. Now for the theorem: the Virial Theorem states that if the time average of dG/dt is zero, then the following holds (we use angle brackets ⟨·⟩ to denote time averages):



2⟨T⟩ = - Σi(Fi · ri)



Which may not be surprising. If, however, all the forces can be written as power laws so that the potential is V=arn (with r the inter-particle separation), then



2⟨T⟩ = n⟨V⟩



Which is pretty good to know! (Here, V is the total kinetic energy of the particles in the system, not the potential function V=arn.) For an inverse square law (like the gravitational or Coulomb forces), F∝1/r2 ⇒ V∝1/r, so 2⟨T⟩ = -⟨V⟩.



Try it out on a simple harmonic oscillator (like a mass on a spring with no gravity) to see for yourself. The potential V ∝ kx², so it should be the case that the time average of the potential energy is equal to the time average of the kinetic energy (n=2 matches the coefficient in 2⟨T⟩). Indeed, if x = A sin( √[k/m] · t ), then v = A√[k/m] cos( √[k/m] · t ); then x2 ∝ sin² and v² ∝ cos², and the time averages (over an integral number of periods) of sine squared and cosine squared are both ½. Thus the Virial theorem reduces to



2 · ½m·(A²k/2m) = 2 · ½k(A²/2)



Which is easily verified. This doesn’t tell us much about the simple harmonic oscillator; in fact, we had to find the equations of motion before we could even use the theorem! (Try plugging in the force term F=-kx in the first form of the Virial theorem, without assuming that the potential is polynomial, and verify that the result is the same). But the theorem scales to much larger systems where finding the equations of motion is impossible (unless you want to solve an Avogadro’s number of differential equations!), and just knowing the potential energy of particle interactions in such systems can tell us a lot about the total energy or temperature of the ensemble.

The Virial Theorem

In the transition from classical to statistical mechanics, are there familiar quantities that remain constant? The Virial theorem defines a law for how the total kinetic energy of a system behaves under the right conditions, and is equally valid for a one particle system or a mole of particles.

Rudolf Clausius, the man responsible for the first mathematical treatment of entropy and for one of the classic statements of the second law of thermodynamics, defined a quantity G (now called the Virial of Clausius):

G ≡ Σi(pi · ri)

Where the sum is taken over all the particles in a system. You may want to satisfy yourself (it’s a short derivation) that taking the time derivative gives:

dG/dt = 2T + Σi(Fi · ri)

Where T is the total kinetic energy of the system (Σ  ½mv2) and dp/dt = F. Now for the theorem: the Virial Theorem states that if the time average of dG/dt is zero, then the following holds (we use angle brackets ⟨·⟩ to denote time averages):

2⟨T⟩ = - Σi(Fi · ri)

Which may not be surprising. If, however, all the forces can be written as power laws so that the potential is V=arn (with r the inter-particle separation), then

2⟨T⟩ = n⟨V⟩

Which is pretty good to know! (Here, V is the total kinetic energy of the particles in the system, not the potential function V=arn.) For an inverse square law (like the gravitational or Coulomb forces), F∝1/r2 ⇒ V∝1/r, so 2⟨T⟩ = -⟨V⟩.

Try it out on a simple harmonic oscillator (like a mass on a spring with no gravity) to see for yourself. The potential Vkx², so it should be the case that the time average of the potential energy is equal to the time average of the kinetic energy (n=2 matches the coefficient in 2⟨T⟩). Indeed, if x = A sin( √[k/m] · t ), then v = A√[k/m] cos( √[k/m] · t ); then x2 ∝ sin² and v² ∝ cos², and the time averages (over an integral number of periods) of sine squared and cosine squared are both ½. Thus the Virial theorem reduces to

2 · ½m·(A²k/2m) = 2 · ½k(A²/2)

Which is easily verified. This doesn’t tell us much about the simple harmonic oscillator; in fact, we had to find the equations of motion before we could even use the theorem! (Try plugging in the force term F=-kx in the first form of the Virial theorem, without assuming that the potential is polynomial, and verify that the result is the same). But the theorem scales to much larger systems where finding the equations of motion is impossible (unless you want to solve an Avogadro’s number of differential equations!), and just knowing the potential energy of particle interactions in such systems can tell us a lot about the total energy or temperature of the ensemble.

Tuesday, July 26, 2011
There are 9 types of energy:
chemical energy - energy stored in fuel (ie. food) which is released when chemical reactions take place
kinetic energy - energy of a moving object
gravitational potential energy (GPE) - energy an object contains due to its position
elastic (strain) potential energy - energy stored in an object that is being stretched, squashed, twisted, you name it!
electrical energy - energy transferred by an electric current
thermal (heat) energy - energy of an object due to its temperature. This is partly because of the random kinetic energy of the particles of the object.
nuclear energy - energy stored in an atom’s nucleus
light energy - energy transferred through waves and light particles (photons)
sound energy - energy transferred via sound waves
The Law of Conservation of Energy states that energy cannot be created or destroyed; however, it can be transformed or transferred.
Machines transfer energy for a purpose. The only problem is that not all energy transferred is useful to us because it doesn’t always go where we want it in the form we wanted it in.
Energy Supplied   = Useful Energy Delivered + Energy Wasted

                        Useful Energy Output
Device Efficiency = ----------------------------
                      Energy Supplied as Input

Efficiency can be visually represented by a Sankey Diagram (pictured above). No machine can be more than 100% efficient because we can never get more energy from a machine than we put into it. An inventor in 19th century America claimed to have made a machine that gave out useful energy without energy being supplied to it. It was later discovered he had a partner pedaling away beneath the ground to supply it with energy!

There are 9 types of energy:

The Law of Conservation of Energy states that energy cannot be created or destroyed; however, it can be transformed or transferred.

Machines transfer energy for a purpose. The only problem is that not all energy transferred is useful to us because it doesn’t always go where we want it in the form we wanted it in.

Energy Supplied   = Useful Energy Delivered + Energy Wasted

                        Useful Energy Output
Device Efficiency = ----------------------------
                      Energy Supplied as Input

Efficiency can be visually represented by a Sankey Diagram (pictured above). No machine can be more than 100% efficient because we can never get more energy from a machine than we put into it. An inventor in 19th century America claimed to have made a machine that gave out useful energy without energy being supplied to it. It was later discovered he had a partner pedaling away beneath the ground to supply it with energy!