Friday, February 27, 2015

Episode 027: 27 Triangular Facts and Factoids You Must Know!

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Episode 27 Ladies and Gentlemen. Check it. I think I have said this before, but I really can't say it enough: I think the most daunting aspect of physics is that the foundations required to truly understand what's going on are never really solidly there, and so I am going to great lengths to make sure everything is going to be firmly in place before we move forward. One critical item that is glossed over time and time again in physics class is the importance of triangles and certain relations of triangles that are just accepted as known. Most of my friends in Physics I who were struggling weren't struggling because they were dumb. They would study and work very hard. They would go to office hours. Then when they revealed they were stuck, many times the piece of information they were missing was something that they never knew, and wasn't related in the textbook because it was assumed that the student walked in with that information! And many times, that information had to do with triangles and geometry. So here, for ep 27, are 27 must know facts about triangles.

0) Very simple but it needs to be stated: A triangle is formed by three lines and has three angles. It has three sides. Great!

1) The sum of the angles in any triangle are equal to 180 degrees (or pi radians).

2) Triangles come on four flavors, as far as we're concerned: Scalene, Equilateral (Equiangular), Isoceles, Right Triangles.

3) Scalene Triangles- These comprise most of the triangles out there. They have nothing particular properties about them (other than the 180 degree angle sum) that can be taken advantage of.

4) Isosceles Triangle- A triangle with minimum two equal sides. This gives several unique properties:

5) An Isoceles Triangle has minimum of two equal angles.

6) Any Isoceles Triangle can be broken into two right triangles by drawing a line bisecting the unequal angle. (More on this later)

7) Equilateral Triangle- An equiangular triangle or equilateral triangle has all sides of equal length and all angles at 60 degrees. This is calculated by dividing the total angle sum, 180, by three angles.

8) All equilateral triangles are isosceles, however not all isosceles triangles are equiangular. Why?

9) Right Triangle- Right triangles have one right angle, meaning an angle of 90 degrees. These types of triangles, for physicists, are likely the most interesting and have highly useful properties. This is generally because our coordinate systems are all orthogonal, aka all exist 90 degrees to each other.

10) The side which does not make an angle with the right angle is referred to as the hypotenuse.

11) The sum of the non-right angles in a right triangle are always equal to 90 degrees. If you know one non-right angle of a right triangle, calculating the other is simple.

12) Any side of a right triangle which is not the hypotenuse is referred to as a leg.

13) For a given angle, we can label the two legs with respect to this angle: the leg which forms the angle along with the hypotenuse is referred to as the adjacent side.

14) The side which DOES NOT form the angle is referred to as the opposite side.

15) Either leg can be the opposite or adjacent side depending on which angle you are referring to.

16) The lengths of the two legs of the triangle can be related to the hypotenuse by the pythagorean theorem: a^2 + b^2 = c^2 where a and b are the legs and c is the hypotenuse.

17) The angles of triangle can be calculated if specific leg lengths are known. Trigonometric functions sine, cosine, and tangent of the angles are used along with the leg lengths to calculate angle values.

18) To calculate the asymmetric angle of an isosceles triangle, it is easy to use right triangle rules with the bisected triangle to find "half the angle", and then double this value.

19) Special Right Triangle #1: 3-4-5 triangle: This triangle has sides which are proportional to the lengths 3-4-5. This is an example of a Pythagorean Triple, where all three sides of the triangle have whole number values. This makes for easy calculation if two of the sides are known.

20) Special Right Triangle #2: 45-45-90 Triangle: This triangle is an isosceles right triangle which has angles measuring 45, 45 and 90. In this type of triangle the sides share the proportion 1:1:sqrt(2). This is useful if the length of one of the legs of the triangle is known. In this triangle the lengths of both legs are equal.

21) Special Right Triangle #2: 45-45-90 Triangle Factoid #1: Either leg of the triangle is equal to one half the length of the hypotenuse multiplied by the square root of two. The one-half is due to keeping the radical value in the numerator.

22) Special Right Triangle #2: 45-45-90 Triangle Factoid #2: The hypotenuse of the right triangle is equal to the length of either leg multiplied by the square root of two.

23) Special Right Triangle #3: 30-60-90 Triangle: This triangle has angles measuring 30, 60 and 90. The sides of the triangle share the proportion 1:sqrt(3):2, where 2 is the length of the hypotenuse. This is useful if the length of one of the sides of the triangle is known.

24) Special Right Triangle #3: 30-60-90 Triangle Factoid #1: The length of the shorter leg is always equal to one half of the length of the hypotenuse.

25) Special Right Triangle #3: 30-60-90 Triangle Factoid #2: The length of the longer leg is always equal to one half of the length of the hypotenuse multiplied by the square root of three.

26) Special Right Triangle #3: 30-60-90 Triangle Factoid #3: The length of the longer leg is always equal to the shorter leg length multiplied by the square root of three.

27) Vector quantities can be described as related to their "component" vectors, which form a right triangle for coordinate axes which are orthogonal.

Tuesday, February 3, 2015

Episode 026: 8 Ideas to Effectively and Productively Use Your Textbook.

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The ol' textbook. It's one of those things that seems so obvious and yet can be so easily cast aside. Included this week are 8 HOT tips to use your textbook. Included below is an image version that you can print out and hang above your bed at night. Have fun!

Read It!- I know this may sound like a very simple thing but I can't tell you how many classes I would only use the textbook as a last resort when what I learned in class wasn't enough. Getting into the habit of reading your textbook is a simple, but major step forward.

Read The Night Before Lecture- This is a great way to get a preview of your next class. If your professor is following the textbook closely you will be in great shape the next day. Also a great way to relax before bedtime! Let your subconscious crank on the new information and then use class to reinforce it.

Don't Get Overwhelmed- Sometimes the text is dry, but also filled with a lot of information in a very short amount of text. Take it slow, reread it if necessary, and don't give up. If you even get a piece of it during the read, you will be able to grab the rest during lecture. You'll also know exactly which questions to ask.

Get a Highlighter- Don't highlight the whole book! Use your highlighter wisely and for absolutely delicious information only. It will save you tons of time when trying to find those passages later when you're doing your homework.

Dig for Gold- Sample Problems and Step-By-Steps are gold just waiting to be unearthed. Most of the homework problems are variants or will easily fit into step-by-step procedures.

Look at Your Homework First- Many times your homework will be assigned early. Even if you don't start early, knowing what you have to do is great at being able to pinpoint sections of the textbook that will help you later.

Review what you Read- When you're feeling saturated, take a break and think about what you just read. It will be a good rehash, and force your brain to solidify those memory pathways for good retention!

Start Making Your Cheat Sheet- Do not take copious notes. If you find phenomenal information that you think would save your life on a test, take a sheet that you can put in the book and jot down a very short amount of information. You'll be starting your cheat sheets and making intelligent notes instead of encyclopedias worth of notes.

Physics Phrydays 004: The Elegant Universe Chapter 2- Space, Time and the Eye of the Beholder

When we last left off, Brian Greene was telling us about the components of the universe, Atoms, made of Protons and Neutrons, which are made of Quarks, which are made of Strings. Setting aside the idea of strings for a moment, and without telling us why, we dive into Relativity, which is probably going to be critical to our understanding of things later on.

One critical component that must be understood is the nature of light. Light is "actually" an electromagnetic wave. It oscillates sinusoidally with an electrical component which is perpendicular to the magnetic component. It also propagates at light speed. And it does not slow down. This is not entirely true, as light in different media travels at different speeds, but what is very important to understand is that the speed in any medium does not attenuate, like a car when you take your foot off the gas. Light just keeps on rolling, rolling, rolling along.

Something that I have mentioned quite a bit in the podcast, probably akin to beating a dead horse, is that reference frames are so critical. You need your coordinate axis. You need to know what you are measuring WITH RESPECT TO. Greene illustrates very well why this is so necessary with a couple of classic relativity examples.

First we get the George and Gracie example. Imagine two astronauts hanging out in space. There is NOTHING else around for zillions of light years. George is hanging out and he sees a flashing light in the distance. This is Miss Gracie. She zooms past and as far as George is concerned he was sitting still and this madam was in motion. Now, if you consider it from Gracie's perspective, it's pretty much the same thing. Gracie is sitting still minding her own business and Mr. George zooms by.

The major takeaway here is that different reference frames will actually measure different events. This was a major change in the way that the universe was thought of.