Stupid pulley question I started self-studying physics and it's been 15 years, since I took basic mechanics in college. It seems to me that I'm unable to figure out how to rigorously derive from Newton's laws how a basic pulley problem works. The answer is obvious and my book jumps to this conclusion. Here's a simplified version of the problem I'm solving.

Assume that we have a pulley hanging from a string from the roof
  (string connected to the center of the pulley). Over the pulley we
  have a string and in both ends there are blocks of equal mass, say,
  $m$, so the system is balanced. I can't figure out how to derive the obvious result that if the
  tension of the string over the pulley is $T$, then the tension of the
  string attached to the roof is $2T$. Assume the strings and the pulley have no mass.

It's clear that the tension on the string over the pulley is $T=mg$ and if we reduce the whole pulley system to a pointmass, then it has mass $2m$, so the tension on the string attached to the roof is $2mg=2T$.
The problem is that the argument of reducing the system to a pointmass is not something that is an obvious consequence of Newton's laws. This issue comes up in a problem in the 2nd chapter of Kleppner and Kolenkow, so it's before things like torque and whatnot have been introduced.
Anyone care to explain how you could derive this from just using Newton's 3rd law and setting up some basic constraints?
 A: This isn't a stupid question at all.  Truly understanding even the most basic mechanics requires conceptual subtlety that goes well beyond what would be appropriate for an introductory course or two, so a necessary part of teaching such courses is making leaps like the one you mention and pretending that they're obvious.
The primitive objects in Newtonian mechanics are point masses - these are the things to which Newton's laws apply directly.  Extending Newton's laws to rigid bodies is not trivial - the results are sometimes referred to as Euler's laws, and require the introduction of several constraints and assumptions to derive.
The first of Euler's laws says

In an inertial frame the time rate of change of linear momentum $\mathbf p$ of an arbitrary portion of a continuous body is equal to the total applied force $\mathbf F$ acting on that portion, and it is expressed as 
  $$\frac{d}{dt} \int \rho \mathbf v dV = \int_S \mathbf t\  dS + \int_V \mathbf b \rho\ dV$$
  where $\rho$ is the mass density of the body, $\mathbf t$ is the "surface traction" (dimensions of force per unit area) exerted at the surface of the body, and $\mathbf b$ is the "body force per unit mass" (dimensions of force per unit mass, i.e. acceleration) exerted on the body.

If you are not familiar with calculus, you can simply say that the acceleration of the center of mass of a body is equal to the total force exerted on that part of the body. 
This is sufficient to answer your question.  The extended body can be thought of as the pulley plus the string across it plus the two masses on the ends of that string.  Because everything is stationary, the left hand side of that equation is equal to zero.  The total force exerted on the body can be split up into the tension from the string attached to the ceiling and the total gravitational force pulling the system downward.  Because they must sum to zero, it follows that the tension in the upper string is equal in magnitude to the total weight of everything hanging below it.
Note, by the way, that if the two masses were accelerating (i.e. if one were heavier than the other) this would no longer be true.
EDIT: In the general case, I believe the tension in the string would be
$$ T = \frac{2(m_1^2+m_2)^2}{m_1+m_2}$$
which reproduces the given result if the masses are equal.
