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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.
A fair point, because it reinforces the reason we define "g" to be a positive number. HIdden negative signs are extremely dangerous. That said, a few nit picks:
ReplyDeleteA scalar can be negative, whereas the magnitude of a real vector is a scalar that cannot be negative.
I agree that choosing a convenient coordinate system is both your right and a useful technique. Here, I have in mind a large number of two-body problems.
The error is not in the choice of coordinate system, but in not respecting the coordinate system when writing down the knowns. (I see that as an important learning opportunity.) It needed to be clearly stated that the symbol a represents a negative number in one of them and a positive number in the other. The confusion in the question thus reflects what can be a long-standing problem shared by anyone (regardless of how much advanced math they know) who has not fully absorbed the idea that adding a negative number is the same as subtracting a positive one. That is what leads to T being smaller in both instances.
IMO, this needs to be addressed early because other symbols, particularly x and v, can take on negative values that are also hidden until computed and written down. The technique that some students of mine picked up in high school is to change coordinate systems in mid calculation to avoid negative numbers for v and/or avoid having to solve a quadratic equation, a technique that is very slow to execute.