Mail Time: "Start in the direction of acceleration" when writing Newton's Second Law.

Consider two identical blocks of mass m, each attached to a string that pulls up.  Both blocks are moving down and speeding up, but block 1 has a smaller magnitude of acceleration than block 2.  Which block experiences greater tension?  In one class, confusion ensued.  Paraphrasing the post to the PGP-secure email group:

The free body diagram has tension T up and mg down.  But the answer to "which block experiences greater tension" depends on the coordinate system we choose.  If up is positive, the tension comes out to T = ma + mg, which means block 2's greater acceleration leads to bigger tension.  But if down is positive, the tension is T = mg - ma.  That means block 2's greater acceleration gives a smaller tension.  The choice of coordinate system isn't supposed to affect the physical world!  AARRGH!!! 

Okay, it's absolutely correct to recognize that the tension rankings should not depend on the coordinate system you use.  I think it’s easiest, then, to think in terms of directions without reference to positive and negatives.  Double and triple negative signs - or math errors due to dropped negative signs - are probably what’s causing the issue here.

The acceleration of the block is *downward* - the block is moving down and speeding up, so acceleration is in the direction of motion.  Therefore the forces acting on the block are unbalanced downward: the down forces are greater than the up forces.  

Rather than define a direction as positive in Newton’s second law, instead I suggest teaching your students to “start in the direction of acceleration.” Write “(down forces) - (up forces) = ma”.  Here that’s mg - T = ma. *

Now, you can solve for T, to find T = mg - ma.  Every value in this equation is just a number.  (In mathematical terms, every variable here either represents a scalar, or just the magnitude of a vector quantity.  It's straightforward to do semiquantitative analysis on the equation: as acceleration gets smaller - closer to zero - the tension gets larger, approaching mg.  Block 1 experiences greater tension.

By not defining a coordinate system to begin with, you’ll not run into trouble deciding whether acceleration should be plus or minus.  And your students are less likely to screw up a problem due to mathematical issues.

Hope this helps...

* If in a different problem the blocks were to have an upward acceleration, then the forces would be unbalanced upwards.  We’d start in the direction of acceleration to write T - mg = ma.