pizzzatime:



BBC: HORIZON :: WHO’S AFRAID OF A BIG BLACK HOLE?




The Clear Science Staff has never liked this description of relativity, because it literally uses gravity as a metaphor to explain gravity. There are two ways to think of force fields (like gravity or electrostatics): either there is a field of forces at every point in space, or there’s no field but space is bent and distorted in an equivalent way.
There is a really great book about this by science fiction writer Rudy Rucker (or Rudolf von Bitter Rucker) called Geometry, Relativity and the Fourth Dimension. 

pizzzatime:

The Clear Science Staff has never liked this description of relativity, because it literally uses gravity as a metaphor to explain gravity. There are two ways to think of force fields (like gravity or electrostatics): either there is a field of forces at every point in space, or there’s no field but space is bent and distorted in an equivalent way.

There is a really great book about this by science fiction writer Rudy Rucker (or Rudolf von Bitter Rucker) called Geometry, Relativity and the Fourth Dimension

(Source: pizzzatime, via freshphotons)

Carbon fixation starts with carbon dioxide, which is a single carbon by itself. (It’s got some oxygens bonded to it, but it’s just one carbon atom!) The process of fixation ends with carbons bonded to other carbons. Basically you can think of fixation as stitching carbons together.
Do you realize how important this is? It’s not like it’s super easy to get carbons bonded to one another. They would much rather be by themselves with some oxygens, as CO2. Scientifically this is because of the free energy of these compounds, but we won’t go into that for now. Carbons bonded to other carbons are technically forms of reduced carbon. Reduced carbon is super useful.

Carbon fixation starts with carbon dioxide, which is a single carbon by itself. (It’s got some oxygens bonded to it, but it’s just one carbon atom!) The process of fixation ends with carbons bonded to other carbons. Basically you can think of fixation as stitching carbons together.

Do you realize how important this is? It’s not like it’s super easy to get carbons bonded to one another. They would much rather be by themselves with some oxygens, as CO2. Scientifically this is because of the free energy of these compounds, but we won’t go into that for now. Carbons bonded to other carbons are technically forms of reduced carbon. Reduced carbon is super useful.

If you Clear Scientists want to hear about standing waves and harmonics, how about checking out our series on:
Guitar fretboard physics
Harmonics on a guitar
(It doesn’t have to do with harmonics, but we also explain how an electric guitar works even though you don’t ever plug it into an outlet. Electric guitars actually make their own electricity when you strum them.)

If you Clear Scientists want to hear about standing waves and harmonics, how about checking out our series on:

(It doesn’t have to do with harmonics, but we also explain how an electric guitar works even though you don’t ever plug it into an outlet. Electric guitars actually make their own electricity when you strum them.)

Trying to find the temperature decay from a flame, we came up with two equations:

  1. a heat flux balance, which we derived
  2. Fourier’s law, which we said was more or less a fundamental law

Now, we combine them to get a differential equation (step 1). Step by step we work through eliminating the derivatives with integration. At the end we get an answer with two constants C1 and C2 in it (step 7). This always happens solving differential equations because each integration produces a constant. Then we take the boundary conditions we specified at the top and solve for the constants (step 8). The answer is linear (step 9)! So there’s the answer: the temperature decay is linear.

But there are a couple things to consider here: first this is one-dimensional. We did that to make the math easy. With a real flame, the heat could go in any direction outward, basically in a spherical shape. Also we’ve specified that 20 cm away from the flame the temperature is 20 °C. This is often the case in the real world, because air is free to move around, and some distance from a flame there will always be cool air to be a heat sink. Problems like this can get really complicated if we want to calculate that distance from first principles, but it can be done!

Finding the temperature decay from a heat source (like a flame) got us talking about heat flux. Heat flux is the movement of heat, and heat is going to flux away from the flame. Heat moves from high temperatures to low temperatures, and wherever heat goes it increases the temperature.
We called heat flux in the x-direction qx. Let’s draw a little box and call it a “system" and do what’s called a "heat flux balance" for the system.
In English: The heat flux into the box equals the heat flux out of the box.
In math: (qx at x) minus (qx at x+Δx) equals zero. Now divide both sides by Δx. Now take the limit as Δx goes to zero, which means the system width becomes infinitesimally small.
When you start saying “infinitesimal" you know you’re doing calculus. This is the definition of a derivative, and our balance ends up with “the negative derivative of heat flux in the x-direction equals zero.”
If you translate 3 back to English it says “heat flux is the same at every value of x.”

Finding the temperature decay from a heat source (like a flame) got us talking about heat flux. Heat flux is the movement of heat, and heat is going to flux away from the flame. Heat moves from high temperatures to low temperatures, and wherever heat goes it increases the temperature.

We called heat flux in the x-direction qx. Let’s draw a little box and call it a “system" and do what’s called a "heat flux balance" for the system.

  1. In English: The heat flux into the box equals the heat flux out of the box.
  2. In math: (qx at x) minus (qx at x+Δx) equals zero. Now divide both sides by Δx. Now take the limit as Δx goes to zero, which means the system width becomes infinitesimally small.
  3. When you start saying “infinitesimal" you know you’re doing calculus. This is the definition of a derivative, and our balance ends up with “the negative derivative of heat flux in the x-direction equals zero.”

If you translate 3 back to English it says “heat flux is the same at every value of x.”

What we’re talking about right now is: What’s the temperature decay rate from a hot point like a flame? To get there we stopped and talked about calculus a little bit though, because to do the answer justice we need some calculus. Stay tuned, we’ll be back on the case next week.

What we’re talking about right now is: What’s the temperature decay rate from a hot point like a flame? To get there we stopped and talked about calculus a little bit though, because to do the answer justice we need some calculus. Stay tuned, we’ll be back on the case next week.

We wondered what a derivative is. Imagine you have a graph with temperature on the y-axis and x on the x-axis. If you pick two points on the graph you can calculate the difference in their y values and the difference in their x values. Dividing those, you would get ΔT/Δx.
In the top-left graph we pick two points far apart. Going from the first point to the second we move 3.1 spaces down on the y-axis, so that ΔT is -3.1. We move 5.5 spaces on the x-axis so that Δx is 5.5. Doing the math it’s -0.56.
But look, if we pick different points we get different values. In the top-right we get -1.67, and in the bottom-left we get -0.36. It depends on what two points we pick.
Now this is a derivative: what if we say the two points we pick are zero distance apart so essentially they are the same point? That is dT/dx, shown in the bottom-right. Each point on the graph will have a different dT/dx value, which is the derivative at that point.
This is now calculus btw, because we talked about two points zero distance apart. (Or an “infinitesimal distance apart” which means infinitely close together.)

We wondered what a derivative is. Imagine you have a graph with temperature on the y-axis and x on the x-axis. If you pick two points on the graph you can calculate the difference in their y values and the difference in their x values. Dividing those, you would get ΔTx.

In the top-left graph we pick two points far apart. Going from the first point to the second we move 3.1 spaces down on the y-axis, so that ΔT is -3.1. We move 5.5 spaces on the x-axis so that Δx is 5.5. Doing the math it’s -0.56.

But look, if we pick different points we get different values. In the top-right we get -1.67, and in the bottom-left we get -0.36. It depends on what two points we pick.

Now this is a derivative: what if we say the two points we pick are zero distance apart so essentially they are the same point? That is dT/dx, shown in the bottom-right. Each point on the graph will have a different dT/dx value, which is the derivative at that point.

This is now calculus btw, because we talked about two points zero distance apart. (Or an “infinitesimal distance apart” which means infinitely close together.)

Asking about the temperature near a hot flame we brought up an important equation called Fourier’s law. The heat flux (q) away from a flame is a constant (k) times the negative of the temperature gradient. And we symbolized the temperature gradient with an upside down triangle in front of T.
That upside down triangle is called “del” and if we’re only worried about one dimension (the left-right dimension in the picture, which we’ll call the x-direction), this “del T” is the derivative of temperature with respect to that dimension. You write it dT/dx. You usually say it “dee-T dee-x” or “dee-T by dee-x.”
If you know calculus, you’ll recognize that is what we’re doing. It’s not really that complicated a concept though. So: What is a derivative, really?

Asking about the temperature near a hot flame we brought up an important equation called Fourier’s law. The heat flux (q) away from a flame is a constant (k) times the negative of the temperature gradient. And we symbolized the temperature gradient with an upside down triangle in front of T.

That upside down triangle is called “del” and if we’re only worried about one dimension (the left-right dimension in the picture, which we’ll call the x-direction), this “del T” is the derivative of temperature with respect to that dimension. You write it dT/dx. You usually say it “dee-T dee-x” or “dee-T by dee-x.”

If you know calculus, you’ll recognize that is what we’re doing. It’s not really that complicated a concept though. So: What is a derivative, really?

Fourier’s law was first formulated by Jean Baptiste Joseph Fourier, whose name is pronounced like “Foo-ree-ay.” The Fourier transform and the Fourier series are important concepts in math.
(Awesome animated math gifs if you click those, FYI.) 

Fourier’s law was first formulated by Jean Baptiste Joseph Fourier, whose name is pronounced like “Foo-ree-ay.” The Fourier transform and the Fourier series are important concepts in math.

(Awesome animated math gifs if you click those, FYI.) 

The Clear Science staff was going to answer the question "Is there a decay rate in heat at distance from a flame/heat source?" To do that let’s consider one way that heat transports from one place to another: conduction.
Heat is energy. Say we have a flame on the left and no flame on the right. The flame is there because some chemical reaction is happening: chemical bonds are breaking and their energy is being liberated. Because of this the temperature of the flame is high, like 1500 degrees. On the right temperature is only room temperature or 20 degrees.
Heat moves by conduction from high temperatures to low ones. This is a basic property of the universe, and it is described by Fourier’s law. Written above in “math language,” what it says in English is "heat flux is proportional to the negative of the temperature gradient." Or: heat fluxes from high temp to low.

The Clear Science staff was going to answer the question "Is there a decay rate in heat at distance from a flame/heat source?" To do that let’s consider one way that heat transports from one place to another: conduction.

Heat is energy. Say we have a flame on the left and no flame on the right. The flame is there because some chemical reaction is happening: chemical bonds are breaking and their energy is being liberated. Because of this the temperature of the flame is high, like 1500 degrees. On the right temperature is only room temperature or 20 degrees.

Heat moves by conduction from high temperatures to low ones. This is a basic property of the universe, and it is described by Fourier’s law. Written above in “math language,” what it says in English is "heat flux is proportional to the negative of the temperature gradient." Or: heat fluxes from high temp to low.