Category Archives: Physics

What does it mean to unify physics?

My son likes physics, and knows that the “Holy Grail” of physics is to unify the models for the different types of physical forces into one. He knows, for example, that while quantum mechanics successfully describes the very small and relativity the very large, they can’t be used together to describe what goes on in a black hole or in a “big bang,” situations where tiny distances and huge masses coexist.

But he wanted to know what it actually means to unify two models. So I pulled a physics book off his shelf and rummaged around in the section on relativity for a formula I remembered seeing oh-so-many decades ago, and told him the following.

Imagine being out in space when a rocket zooming by at velocity v1 (relative to us) shoots a projectile in its direction of travel at velocity v2 (relative to the ship). Classical mechanics tells us that the velocity of the projectile relative to us is given by the equation

v = v1 + v2

But eventually it was discovered that nothing can go faster than the speed of light. The equation above works fine for low velocities, but fails when v1 and v2 are close to the speed of light (for which we use the symbol “c”), because v can never exceed c.

Einstein (well, let’s not forget Larmor and Lorentz) eventually gave us this equation, which fixes the problem:

v = ( v1 + v2 ) / ( 1 + v1*v2/c^2 )

When v1 and v2 are small relative to c, the bottom becomes very close to 1, so the old equation from classical mechanics gives an answer that is correct to a very high degree of accuracy. But when v1 and v2 are close to c, the bottom becomes significantly more than 1, enough to prevent v from reaching c. This equation “unified” classical mechanics with the fact that nothing can travel faster than light.

The same thing will happen in upcoming unifications. We have equations which very accurately describe the physical universe under certain conditions; whatever the equations of the unified model ultimately look like (and I have not the slightest clue about that), they will have to effectively reduce to the equations we have now under the right conditions. The new equations will be more general in that they will work under more (perhaps all) circumstances, because of terms which we don’t know about yet. These new terms will disappear when they need to (effectively giving us our old equations) and play a part when they need to (under circumstances that the current equations can’t handle).

He liked that explanation.

How much stronger is a 0.9mm pencil lead than a 0.5mm?

I don’t know why 0.5mm pencils are more popular than 0.7mm and 0.9mm. I had 0.5mm pencils around the house, but when I saw my son break the lead three times in 15 seconds I decided to try 0.7mm. Much less breakage. At the time I didn’t even know that you could get 0.9mm pencils, but when I found out, I got one. I was afraid that they would be too thick, but in fact the lines they produce doesn’t really seem much thicker — it’s just smoother to write them. That’s probably because the worn surface of the lead is curved, and for the same amount of pressure a 0.9mm lead will penetrate less deeply into the paper. That reduces both the spot size and the drag of the pencil.

The big win, though, is that 0.9mm leads are much less prone to breakage than 0.5mm. I’ll spare you the math, but at a first approximation the strength of the lead goes up with the cube of the diameter, so according to the simple model I used (I make no claims to either accuracy or precision) it takes something like 6 times as much force to break a 0.9mm lead as a 0.5mm. Also, because the 0.9mm lead is not making as sharp a trough in the paper, there is less perpendicular force on the lead. I’ve been using 0.9mm pencils for a year or so now, and I have yet to break a lead.  This diagram shows their relative sizes:

Other advantages to using thicker lead:

– You don’t have to advance the lead as often.
– You can make a bolder line, if you want to. I do a lot of underlining and quick diagrams, and bolder lines work well for both.
– The mechanism seems less finicky. I used to have to disassemble 0.5mm pencils from time to time, but I never do that anymore.
– Lines are more easily erasable because the lead hasn’t made so sharp a trough in the paper.

Now I can understand that for somebody doing drafting, being able to make a really fine line is important. Such folk may even need a 0.3mm pencil. But if you are just doing homework or taking notes, you might want to try a 0.9mm.

By the way, I think it would be a really cool high school science experiment to empirically determine the relative strengths of leads of different diameters.

And you might be curious why the strength goes up with the cube rather than the square of the diameter (or radius). Clearly the cross-section of the lead, which must be pulled apart, is a disk, the area of which increases with the square of the diameter. If you were just trying to pull the lead apart by pulling the two ends in opposite directions, the strength would indeed increase with the square of the diameter. But while writing, a lead breaks in response to a perpendicular force — effectively making a lever — and the cohesion of the lead has more leverage the farther away it is from the fulcrum. A larger-diameter lead adds surface area mostly on the side furthest from the fulcrum, where it does the most good.