Physics: Theories of Everything (Mapped)

March 7, 2016

An excellent overview of the multitude of active theories and hypotheses–like quantum gravity, string theory–physicists are investigating to try to explain the universe.

From Theories on Everything Mapped

From Theories on Everything Mapped

In the quest for a unified, coherent description of all of nature — a “theory of everything” — physicists have unearthed the taproots linking ever more disparate phenomena. With the law of universal gravitation, Isaac Newton wedded the fall of an apple to the orbits of the planets. Albert Einstein, in his theory of relativity, wove space and time into a single fabric, and showed how apples and planets fall along the fabric’s curves. And today, all known elementary particles plug neatly into a mathematical structure called the Standard Model. But our physical theories remain riddled with disunions, holes and inconsistencies. These are the deep questions that must be answered in pursuit of the theory of everything.

–Natalie Wolchover in Theories of Everything Mapped on Quanta Magazine.

Citing this post: Urbano, L., 2016. Physics: Theories of Everything (Mapped), Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Updated Atom Builder

September 5, 2014

A couple of my students asked for worksheets to practice drawing atoms and electron shells. I updated the Atom Builder app to make sure it works and to make the app embedable.

So now I can ask a student to draw 23Na+ then show the what they should get:

Worksheet

Draw diagrams of the following atoms, showing the number of neutrons, protons, and electrons in shells. See the example above.

  1. 14C: answer.
  2. 32K+: answer.
  3. 18O2-: answer.
  4. 4He2+: answer.
  5. 32P: answer.

I guess the next step is to adapt the app so you can hide the element symbol so student have to figure what element based on the diagram.

Citing this post: Urbano, L., 2014. Updated Atom Builder, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Atom Builder

January 22, 2013

This app lets you drag and drop electrons, protons, and neutrons to create atoms with different charges, elements, and atomic masses. You can also enter the element symbol, charge and atomic mass and it will build the atom for you.

Note, however, it only does the first 20 elements.

Citing this post: Urbano, L., 2013. Atom Builder, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Everything You (N)ever Wanted to Know About Parabolas

March 7, 2012

So that my students could more easily check their answers graphically, I put together a page with a more complete analysis of parabolas (click this link for more details).

Analyzing Parabolas

Standard Form Vertex Form
y = a x2 + b x + c y = a (x - h)2 + k
y = x2 + x +

y = ( x - ) 2 +

Intercepts: Vertex:
Focus: Directrix: Axis:
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Solution by Factoring:

y = x2 x

Converting the forms

The key relationships are the ones to convert from the standard form of the parabolic equation:

         y = a x^2 + b x + c (1)

to the vertex form:

         y = a (x - h)^2 + k (2)

If you multiply out the vertex equation form you get:

         y = a x2 - 2ah x + ah2 + k (3)

When you compare this equation to the standard form of the equation (Equation 1), if you look at the coefficients and the constants, you can see that:

To convert from the vertex to the standard form use:

         a = a (4)
         b = -2ah (5)
         c = ah^2 + k (6)

Going the other way,

To convert from the standard to the vertex form of parabolic equations use:

         a = a (7)
         h = \frac{-b}{2a} (8)
         k = c - ah^2 (9)

Although it is sometimes convenient to let k not depend on coefficients from its own equation:

         k = c - \frac{b^2}{4a} (10)

The Vertex and the Axis

The nice thing about the vertex form of the equation of the parabola is that if you want the find the coordinates of the vertex of the parabola, they're right there in the equation.

Specifically, the vertex is located at the point:

         (x_v, y_v) = (h, k) (11)

The axis of the parabola is the vertical line going through the vertex, so:

The equation for the axis of a parabola is:

         x = h (12)

Focus and Directrix

Finally, it's important to note that the distance (d) from the vertex of the parabola to its focus is given by:

         d = \frac{1}{4a} (13)

Which you can just add d on to the coordinates of the vertex (Equation 11) to get the location of the focus.

         (x_f, y_f) = (x_v, y_v + d)  (14)

The directrix is just the opposite, vertical distance away, so the equation for the directrix is the equation of the horizontal line at:

         y = y_v + d  (15)

References

There are already some excellent parabola references out there including:

Citing this post: Urbano, L., 2012. Everything You (N)ever Wanted to Know About Parabolas, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

3d Molecule of the Month

September 24, 2011

Cyclohexane, from the interactive model on 3Dchem.com.

Molecular models tend to fascinate. As a introduction to the chemistry of elements, students seem to like putting them together, and they tend to enjoy finding out what their molecules are called. You can't beat fitting together molecules by hand as a learning experience, but 3Dchem has a nice collection of interactive, three-dimensional molecules, including molecules of the month.

Periodic spiral of the elements (from 3Dchem.com).

They also have three-dimensional periodic tables showing the sizes of the atoms in the traditional tabular form as well as a spiral.

Periodic Table showing the elements by size.

Citing this post: Urbano, L., 2011. 3d Molecule of the Month, Retrieved July 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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