10228 Haunted House review is coming.
My roommate is thinking of going into school to do research. I am not terribly keen on the idea. So, Pokémon it is. 8D
Also, that awkward moment when you're about ready to check out Christmas presents for siblings on Bricklink and a scheduled downtime occurs. :\
Where E is energy, m is mass, and c is the speed of light. It's a very simple-looking equation with only three parameters, but what does it mean? Well, it means that anything with mass – you, your cat, your house, the Earth – has latent energy stored in it, and the amount of mass determines that latent energy. For an object at rest, this correlates to the rest mass of the object. If an object is moving really fast (near the speed of light) its kinetic energy causes it to actually get heavier, since the object can never actually reach the speed of light (only objects with no rest mass move at the speed of light).
So, if we have an object sitting and doing nothing, and it suddenly glows for a split second, then stops, where did the light come from? Well, light has energy, as we know, so we could calculate the energy of the light that escapes our object. If the light emanates in all directions, then the net kinetic energy of the object is unchanged. But conservation of energy says that energy can neither be created nor destroyed! Have we violated the laws of physics with our weird glowing object? Well, no, because if you were somehow able to weigh the object pre- and post-glow, you would find that the mass of this object is actually slightly less after the light is given off.
But wait! Doesn't conservation of mass say that matter can neither be created nor destroyed? Well, yes, it does say that. So the only way for this to make sense is if the mass is converted into the energy that was emitted. We know that energy can be converted into different forms (electric, mechanical, thermal, etc.), so this must mean that mass is another form of energy that can be converted to and from! Pretty neat, huh?
Minutephysics has a cool video on this with a bit more technicality and pretty pictures of radioactive cats, but this is my text-based explanation simplified.
Another thing that may cross your mind is that this looks very similar to Newton's second law:
So, does Newton's second law equate force with acceleration? Well, no, because in the mass-energy equation, the constant of proportionality, c2, is a universal constant; it is the same for any and all objects in the universe. The mass of an object, however, varies from object to object, and is thus not a fundamental, universal constant, so while these equations are similar and relate two seemingly different entities, they do not conceptually perform the same task.
I also received and played intensely Pokémon White 2, hence my lack of blog posts. I just beat the 8th Gym today (Marlon is weird...) and now have to prepare lecture for Wednesday. :\
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.
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.
Now if only all the trees decided to finally change color to match the rest of the mood.
Gravity is a force between objects/particles proportional to the objects' mass. Newton's universal gravitation looks like this:
Fg = - G m1m2/r2
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
Fg = m a = - G m ME/r2
where ME 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 ME/r2
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 c2
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.
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...
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Hieroglyphs And The Like
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