Tag: Esther Chilton

Conic Sections are the Shapes that Shape Our World

Super fact 80 : A conic section is a shape formed by slicing a cone with a plane. There are four such shapes, circle, ellipse, parabola, and hyperbola. The conic sections universally describe motion under gravity. The orbits of planets around their stars are circles or ellipses, comets fly around space in elliptical orbits, or parabolic or hyperbolic paths. Objects thrown up in the air follow parabolic paths. They are the basis for a huge amount of engineering applications.

Esther’s writing prompt: January 21 : Shapes

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Four cones each shown with a plane section forming a specific conic section. | Conic Sections are the Shapes that Shape Our World
Types of conic sections : circle , ellipse , parabola , hyperbola Shutterstock Asset id: 2377159367 by ProfDesigner

The four conic sections, circle, ellipse, parabola and hyperbola are fundamental and very useful shapes in mathematics, physics and engineering. Well, a circle is a special case of an ellipse, so it is really only three conic sections. The motion of the planets and other stellar objects are described by the conic shapes. Isaac Newton derived his law of gravitation from Kepler’s laws, which describe planetary orbits as ellipses.

The conic sections are all described by second degree equations (quadratic equations) and are in that sense the simplest shapes aside from points and lines. It is important to understand that there is an infinite amount of shapes that are almost conic sections and look like conic sections, but it is the exact mathematical properties of the four conic sections that make them so common in physics, mathematics, nature and engineering.

The picture shows a cone with four planes slicing the cone in four ways. The resulting shapes are circle (red), ellipse (green), parabola (blue), hyperbola (orange).
The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. by Magister Mathematicae, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=18556148

It may not come as a surprise that the circle is a fundamental and important shape, but I believe that the fact that the other conic sections are also fundamental in mathematics, physics and engineering come as a surprise to people outside of the STEM fields. It is a true and an important fact regarding how our world works.

Conic Sections

As mentioned, the conic sections are fundamental shapes that appear in a lot of places in STEM. Below are a few examples.

Parabola

Math function parabola graphics illustration with a dark background. | Conic Sections are the Shapes that Shape Our World
Math function parabola. Shutterstock Asset id: 1628916337 by EleonoraDesigner

A parabola is formed when a plane cuts a cone, so the plane is parallel to a side of the cone. Parabolas are shapes that are roughly U-shaped and described by the equation y = x^2 or more generally by y = ax^2 + bx + c. Parabolas have a so called focus point. See the picture below. If you throw a ball, or any object, up in the air its trajectory will be a parabola (ignoring distortions caused by friction and wind). I should say the parabola you get in this case is upside down. The parabola is important when you design any kind of projectile.

U-shaped parabola with the focus shown. The pciture has an x-axis and a y-axis.
Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward. Picture is from Wikipedia Melikamp, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons.

Antennas shaped like parabolas (in 3D) will direct incoming radiation and waves towards their focus point. If the surface is reflective a light located at the focus point will reflect to create a straight beam. Parabolas are used for radio telescopes, satellite dishes, car headlights, flashlights, solar cookers, solar power plants, water fountains, suspension bridges, business modelling and thousands of engineering applications. Parabolas like circles and the other conic sections shape our modern world (pun intended).

A parabola dish with equipment located at the focus.
Würzburg-Riese radar built by Germany in WW2 had a 7.4 meter (24 foot) dish. From this page. Alan Wilson from Stilton, Peterborough, Cambs, UK, CC BY-SA 2.0 https://creativecommons.org/licenses/by-sa/2.0, via Wikimedia Commons

Ellipse and circle

As mentioned, a circle and an ellipse are conic sections formed by intersecting a plane with a cone. You get a circle when the cuts perpendicular to the cone’s axis (see pictures above) and an ellipse form when the plane intersects the cone at a slant but not slanted so much that it becomes a parabola or a hyperbola. An alternative for an ellipse is that the sum of the distances from any point on the curve to two fixed points (called the foci) is a constant. See the picture below. The two definitions are identical. For a circle the two foci are merged into one point at the center.

The picture shows an ellipse and its two foci points. From the foci points there are lines going to a point P on the ellipse. The length of the two lines are added together and is the sum “2a” no matter where on the ellipse the point P is located. | Conic Sections are the Shapes that Shape Our World
Ellipse: definition by sum of distances to foci. Ag2gaeh, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

There are a lot of real world examples of ellipses. Planets orbit the Sun in elliptical paths. The sun is in one of the foci points. The orbits of other stellar objects and satellites are also elliptical. Charged particles follow elliptical paths within magnetic fields.  Elliptical patterns are observed in the rotation of ocean currents, elliptical models and algorithms are used in medical imaging, computer science and encryption. Also whispering galleries.

Hyperbola

Comets and spacecraft that are not orbiting another body, in other words, they have enough speed to escape the gravitational pull and continue into deep space, will travel along a hyperbola. The boundary of a shockwave from a supersonic jet (a sonic boom) creates a hyperbolic curve on the ground as it moves. The intersection of two sets of concentric ripples in water makes a hyperbola. The light beam from a lamp or flashlight makes an ellipse or an hyperbola on a plane depending on the angle.

Newton’s Law of Gravitation

Johannes was an early 17th century German mathematician who derived three laws that describe how planetary bodies orbit the Sun using the observational data collected by the Danish astronomer Tycho Brahe. The three laws are the following:

  • Planets move in elliptical orbits with the Sun as a focus.
  • A planet covers the same area of space in the same amount of time no matter where it is in its orbit.
  • A planet’s orbital period is proportional to the size of its orbit (its semi-major axis).
Kepler’s three laws are illustrated in a diagram for two planets.
Illustration of Kepler’s laws with two planetary orbits.
The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1.
The shaded areas A1 and A2 are equal and are swept out in equal times by Planet 1’s orbit.
The ratio of Planet 1’s orbit time to Planet 2’s is (a1/a2)^3/2
Hankwang, CC BY-SA 3.0 <http://creativecommons.org/licenses/by-sa/3.0/&gt;, via Wikimedia Commons

Later Isaac Netwon would use Kepler’s three laws to derive his law of gravity. Newton showed that an inverse-square force (gravity) directed toward the sun was necessary to explain the orbits.

My Other Responses to Esther’s Prompts

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Electric Charge is not the only type of Fundamental Charge

Super fact 59 : Most people have heard of electrical charges, positive and negative. However, in nature there are also color charges—red, green, and blue—which are analogous to electric charges. In addition, there are anti-red, anti-green, and anti-blue charges.

Esther’s writing prompt: 10th September : Charge

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As you may know, atoms consist of particles. Electrons surround the nucleus of the atom. The nucleus of the atom is in the middle of the atom and it consists of protons and neutrons. Electrons have a negative charge. Protons have a positive charge. Neutrons do not have an electrical charge. Electrons are so called elementary particles. They are not composed of other particles. Protons and Neutrons, on the other hand, are not elementary particles. They are composite particles consisting of quarks, gluons and quark pairs called mesons.

The picture shows a Hydrogen atom consisting of one proton and one electron, one Carbon atom with six electrons, six protons and six neutrons, an Oxygen atom with eight electrons, eight protons and eight neutrons, and a Nitrogen atom with seven electrons, seven protons and seven neutrons | Electric Charge is not the only type of Fundamental Charge
Four elements with a nucleus and electron shells. The number of electrons, protons, and neutrons is shown. The green particles circling the nucleus are electrons. The red particles in the nucleus (middle) are protons and the blue particles in the nucleus are neutrons. The colors of the particles in this picture have nothing to do with color charges. The four elements are Hydrogen, Carbon, Oxygen, and Nitrogen. There are 118 elements. These elements can combine into millions of different kinds of molecules that make up everything. Asset id: 1555863596 by OSweetNature.

Quarks have electric charges, just like an electron and a positron, which is why a proton has an electric charge, a positive electric charge. However, in addition quarks have something called color charge. Unlike electric charges, which come in two forms, negative and positive, they come in three forms red, green and blue and in anti-red, anti-green, and anti-blue (well six forms actually). I should say that the color charges, red, green and blue, are not real colors. They are just names. Just electric charges are associated with electric forces; color charges are associated with the nuclear strong force. The strong force is even stronger than the electrical force.

If you take an equal amount of positive and negative electric charges you get something that is electrically neutral. If you take an equal amount of red, green and blue you get what is called white, or neutral. If you take an equal amount of red and anti-red you also get white. Any other mix gives you a net color charge.

vector illustration of up and down quarks in proton and neutron on white background. The proton (left) is a red and blue up quark and a green down quark. The neutron is a red and green down quark and a blue up-quark.
The proton and neutron each consist of three quarks. Protons consist of two up quarks and one down quark. Neutrons consist of two down quarks and one up quark. Both protons and neutrons have a net white charge. The yellow squiggly lines are gluons transporting color charge between the quarks. Asset id: 2333679305 by KRPD.

I can add that gluons are elementary particles that in many respects are like photons. Light consists of photons. It is because of the photons that we can see. In addition, the photons transport electrical charge. Photons are massless elementary particles with the intrinsic spin of one, and they belong to a group of elementary particles called Bosons. Gluons transport color charge, and they are massless and have an intrinsic spin of one and belong to the same group of elementary particles called Bosons. Unlike photons, they are stuck inside the nucleus and unlike photons they never get to see the light of day. The pun was intended.

Matter, light, and electrical charges are all part of our daily life. We can touch matter, see light, and we come across electrical charge when we touch something that is charged or when we see lightning. However, we do not come across quarks, gluons, and color charges in our daily life because they are hidden at the center of the atoms. Yet they are fundamental to the existence of matter, of us. We know color charges exist, the existence of color charges is an important fact, and yet it is not a well-known fact and often a big surprise to people. Therefore, I think it is a super fact.

The 118 Elements and the 3,500 Isotopes

There are 118 known elements. Why not 500, or just 4 or 5, like the ancient Greeks believed? Each element is defined by it having a certain number of protons and the same number of electrons if it is to be electrically neutral. The problem with having more than one proton in the nucleus is that protons all carry a positive charge and therefore want to push each other away. Same charges repel and different charges attract. What saves the nucleus from blowing apart are the neutrons and the associated strong nuclear force (protons & neutrons) which is guided by the color charges. The quantum model for electricity is called Quantum electrodynamics or QED. The quantum model for color charges is called Quantum chromodynamics or QCD.

As you add more protons it becomes increasingly more difficult for the nuclear forces (strong and weak) to hold the nucleus together. The positive charge of the protons is pushing too hard. That’s why there are only 118 Elements. Another thing to note is that the number of neutrons does not have to be the same as the number of protons. This means that for each element there are several kinds of so-called isotopes. For example, carbon has six protons and six electrons (if the atom is electrically neutral) but the carbon atom / element can have six neutrons, seven neutrons, or eight neutrons. You call them carbon-12, carbon-13, and carbon-14, where the number represents the number of protons plus the number of neutrons.

The picture shows a Carbon-12 isotope, a Carbon-13 isotope, and a Carbon-14 isotope | Electric Charge is not the only type of Fundamental Charge
Three natural isotopes of Carbon Stock Vector ID: 2063998442 by zizou7
Bohr model representation of the uranium atom, number 92 and symbol U. Conceptual vector illustration of uranium-238 isotope atom, mass number 238 and electron configuration 2, 8, 18, 32, 21, 9, 2.
This is a simplified Bohr model of the Uranium atom. There are 92 little blue balls circling a nucleus in the middle of the atom. Those are electrons. In the nucleus there are 92 protons. Those are the red balls with plus signs. In addition, there is a yellowish smudge around the protons in the nucleus. Those are the neutrons. Depending on the isotope, there are 143 neutrons for U-235, 146 neutrons for U-238 and 142 neutrons for U-234. Shutterstock asset id: 1999370450 by Patricia F. Carvalho

It is the electrons that determine the chemical properties of an element, and therefore isotopes with a different amount of neutrons are chemically identical. However, they are different with respect to properties that relate to he nucleus, such as radioactivity/stability, and of course weight. Also, when atoms combine into molecules their chemical properties change drastically, but again that is due to the rearrangement of the electrons. There are around 3,500 known isotopes, most of them radioactive.

What is a Quark?

To learn more about Protons, Neutrons, Quarks, Gluons, Color Charges, and Quantum Chromodynamics you can watch this 10 minute video below.

Other Physics Related Superfacts

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