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Posted by
Dr. Akano
,
in
Math/Physics,
Life
Oct 13 2012
·
227 views

I had a wonderful time today; I got to see old friends from my undergrad today and went exploring a corn maze; it was a lot of fun.

Also, equation of the day: Newton's Second Law

where F is the net force, a is acceleration, and m is the mass of the object in question. It's such a simple-looking equation, but it contains so much physics. Want to know the path of a free-falling object subject only to the force of gravity? You use this equation. Want to know the attractive force and classical orbit of planets/atoms? You use this equation. Want to know the physics of a car skidding on pavement? You get the idea.

This equation is a staple of physics and is used extensively in intro and classical physics. Newton, you clever devil, you.

So, I've decided to do one of those daily-like blog entries, though I can't guarantee that I'll be able to do this every day (being a busy grad student and all). I figured that, being a physics grad student, math might be one of my stronger suits (next to reviewing LEGO sets), so I'm going to try and share an equation with you and see if I can explain it well enough for people to understand. 8D

Tonight's equation: The wave equation.

This says that the sum of the change in the change in the function, ψ, with respect to the coordinates used to represent it is equal to the inverse square of the speed of the wave,c, modeled by ψ times the change in the change of ψ with respect to time.

This equation is the governing equation for all wave phenomena in our world. Sound waves, light waves, water waves, earthquakes, etc. are governed by this mathematical equation. In one dimension, the wave equation simplifies to

which has the lovely solutions

where A and B are determined by appropriate boundary conditions, and ω/k = c. This equation governs things like vibrations of a string, sound made by an air column in a pipe (like that of an organ, trumpet, or didgeridoo), or even waves created by playing with a slinky. It also governs the resonances of certain optical cavities, such as a laser or Fabry-Perot cavity.

Since waves are one of my favorite physical phenomena, I find it very appropriate to start with this one.

I went for a couple walks yesterday and today, and everything feels perfect. The air is crisp, the hoodie/jeans combo is perfectly comfortable, and the smell of fallen leaves permeates everywhere.

Now if only all the trees decided to finally change color to match the rest of the mood.

Or, so some of my students in my Intro Physics lab think. Hopefully when you read the title you were ready to get your typing fingers ready to disprove me. You probably would have made an argument akin to the following mini-lecture.

Gravity is a force between objects/particles proportional to the objects' mass. Newton's universal gravitation looks like this:

F_{g} = - G m_{1}m_{2}/r^{2}

where G is a proportionality constant, the m's are the masses of the two objects in question, and r is the distance between the two objects. This is why we feel the Earth's gravity affect us, but we don't feel the moon's or sun's gravity affect us. They most definitely influence the Earth (since the sun causes our orbit and the moon causes the tides), but we don't feel the effects of their presence.

So, if we have an object with mass m on Earth in free fall, its equation of motion is determined by

F_{g} = m a = - G m M_{E}/r^{2}

where M_{E} is the mass of the Earth and a is the acceleration of the object. Note that, if we divide both sides by m, we find that

a = - G M_{E}/r^{2}

which means that the acceleration of an object in free fall has nothing to do with the mass of the object. In fact, you can see a video of this on the moon at Wikipedia's Gravitation page that shows Apollo 15 astronaut David Scott dropping a feather and hammer simultaneously. Since there is no air on the moon, the feather is not afloat longer than the hammer, and they fall at the same rate and hit the ground at the same time.

Also, while I said earlier that gravity affects things with mass, it also affects light, which does not have (rest) mass. However, light has energy, and as Einstein showed with his Special Theory of Relativity, energy and mass are equivalent:

E = m c^{2}

So, you can construct the relativistic mass of light, thereby finding the equations that govern the changing of the straight path of light in a gravitational field. Using Einstein's General Theory of Relativity, you can also view the gravitational field as a curvature of spacetime, which influences straight lines to be curved in the space near the massive object, affecting the path of light.

Another interesting thing about mass: objects actually have two different masses associated with them: gravitational mass and inertial mass. Gravitational mass tells you how much an object interacts gravitationally, while inertial mass tells you how much an object resists a change in motion. In other words, more massive objects take more force/energy to alter their paths than objects with less mass. Here's the interesting thing, though: both these masses are equal, even though there really is no physical law stating that they have to be. The only reason we know these masses are equal is because empirical evidence says they are; there is no indication that these two masses are different to an appreciable/statistical extent.

So, if you think that there are no unanswered questions in the realm of physics, you are sorely mistaken.

And it was glorious. I was going to work on my E&M homework, but my body decided to rob me of all focus and initiative. I then ended up in my bed and slept for about three and half hours with no homework progress. Hooray?

Also, I've realized that I haven't finished my Time Cruisers/Twisters reviews, and I kind of want to retake the pictures for them (I have the pictures on my computer of when I first built them and planned to review them, but they're sub par...). Then I'd like to do some more retro (90s) set reviews. I'm thinking of tackling the good ol' Adventurers sets, as I have quite a few of those...

Proto +2 for Premier Membership +1 from Pohuaki for reporting various things in Artwork

Name: Akano Real Name: Forever Shrouded in Mystery Age: 28 Gender: Male Likes: Science, Math, LEGO, Bionicle, Comics, Yellow, Voice Acting, Pixel Art, Video Games Notable Facts: One of the few Comic Veterans still around Has been a LEGO fan since ~1996 Bionicle fan from the beginning Twitter: @akanotoe