It would seem that I haven't posted here in nearly four years. Life has been busy, and further, I find other social media outlets capture my attention more. But I shall try to post here now and then. So what has happened in physics in the last four years? Too much to write! But here are a few things:
• The Large Hadron Collider has pinned down many of the properties of the Higgs boson.
• Alas, there are few hints of anything unexpected, except for a "bump" in the data at six times the mass of the Higgs boson, which might be some new elementary particle. Stay tuned!
• Quantum mechanics has passed all tests thrown at it. There has been great progress in using it to improve communication security and computation--though that is still a ways off.
• There has been a lot of work in constructing materials from the small scale on up.
• Gravitational waves have been directly observed.
The last one was reported just a few months ago. Two giant black holes merged a billion light years away, and scientists were able for the first time to detect the resulting jiggling of spacetime, just as Einstein predicted. It was a stupendous achievement, and opens our ears to a whole new side of the Universe.
Showing posts with label relativity. Show all posts
Showing posts with label relativity. Show all posts
Saturday, May 7, 2016
Science over the last four years
Saturday, March 8, 2008
Why c squared?
[originally posted 2/17/08. Update added to end.]
This post is in response to a comment on my post Physics Factoid: E = m c squared, which asks "Why is the conversion factor c2?" This will be slightly more technical than most of my posts, but should make sense to anyone who has had high school physics.
There are two parts to the question. First, why is the factor something squared? Second, why is that something c, the speed of light?
The answer to the first question is that the factor has to be some speed squared to get the units right. Most people don't value units enough. You can often solve a problem just knowing the units involved, and you can certainly tell if something is askew if the units aren't right. For example, if you ask someone, "what's the area of your living room?" and they answer, "Thirty feet," you know they misheard the question because an area has to be in square-feet.
The units of energy are mass·speed2, for example, kg·m2/s2. The kinetic energy of a mass m moving and velocity v is ½ m v2. [application to space junk]
So any formula writing energy in terms of mass has to involve something with units of speed squared.
But why the speed of light? Well, the theory of relativity is based on the idea that all observers, no matter how fast they are moving relative to you, must observe the same physical laws. And there is only one special speed which they all measure to be the same—the speed of light. In fact, you can show that the theory doesn't work if you try to add a second special speed. A short way of putting it is that in relativity there is only one constant to work with, the speed of light.
Now I could derive the formula for you, or show you it gives the correct kinetic energy in the low-speed limit, but a simple answer is that there is no other speed it could be!
So in summary, the factor has to be a speed squared to get the units right, and the only speed it could be is the speed of light.
[Answer to Comment added 3/8/08]:
Here is an attempt to answer the comment by 'dubiousraves'. I'm going to use some algebra, so stop reading now if you have math phobia! :)
I assume that you accept that the rest energy of any particle is some constant K times its mass, ie
rest E=K m
So we just need to figure out what K is. Further, its kinetic energy (at speeds much smaller than the speed of light) is ½ m v2. Now the total energy of a particle is its rest energy plus its kinetic energy. The theory of relativity says it has to be proportional to the factor called gamma, γ= 1/√(1-v2/c2), which is the same factor that accounts for the length contraction and time dilation. The total energy is thus
total E=γ K m.
When v=0, the particle is at rest and γ=1. In that case the total energy equals the rest energy (when something is at rest, it has no kinetic energy). Now when the speed v is a lot less than the speed of light (which is always true in everyday life), then v2/c2 is much smaller than 1. In that limit we can use the formula 1/√(1-e) ≈1+e/2 (this comes from a Taylor series expansion), for any quantity e that is much smaller than 1. So for small v, γ≈1+½ v2/c2 . If we plug this into the total energy, we get,
total E=γ K m≈(1+½ v2/c2) K m = Km + (K/c2) ½ m v2.
So to get the kinetic energy right, K needs to be c2. As a check, let's plug in K=c2:
rest Energy= K m = m c2,
total Energy = γ K m = γ m c2 ≈ (1+½ v2/c2) m c2 = mc2 + ½ m v2.
I hope that helps.
Wednesday, January 16, 2008
Physics Factoid: E = m c squared
This is probably the most quoted formula. Einstein's equation changed the world. But what does it mean?
I already defined c for you: it is the speed of light.
The only thing you need to know about c in this equation is that it is BIG (30 billion cm/s), and c squared [see Why c squared?], which is c times c, is VERY BIG (900 quintillion cm2/s2). E stands for energy, and m for mass (the amount of matter). So the equation is really
Energy = mass x BIG_NUMBER
Well, what the heck does that mean? It means that matter and energy are really two sides of the same coin called mass-energy. Matter can be converted into energy, and energy can be converted into matter. Before Einstein, it was thought that they were completely separate things. But they are not. They are two different manifestations of the same thing. They are like dollars and yen, except the units are grams and ergs. But there is a VERY steep exchange rate. For every gram of matter, you get 900 quintillion ergs. If you could somehow convert a pound of matter (454 grams) completely into energy, you'd have enough to power the US electrical grid for about 4 days.
Note, you can't actually convert 100% of matter into energy (since we don't have a ready supply of antimatter)—even thermonuclear fusion converts only about 1% of the matter involved into energy.
[confidence level: established, my qualifications: trained]
Monday, January 7, 2008
Physics Factoid: The Speed of Light
I've realized that I'm not always going to have time for a full post. And perhaps some of you would like things in small bites. So I will sometimes publish short posts called factoids. This one is on the speed of light.
The constant c in Einstein's famous equation E=mc2, is the speed of light. c is 186,282 mi/sec, or 299,792 km/sec. That means light could go around the equator of the Earth (in some conduit) about 7 times in a second. On the other hand, the sun is so far away, 93 million miles, that it takes light more than 8 minutes to get here from there.[confidence level: established, my qualifications: trained]
Subscribe to:
Posts (Atom)
