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## 07 August 2012

### Never trust a student with a negative sign

Yesterday's post detailed how I deal carefully with negative signs in six topics where they can't be avoided.  No doubt you are wondering, "how do you avoid negative signs elsewhere?"  Okay, I'll run through some topics in which negative signs *can* be used, but in which I manage to avoid them.

Equilibrium of forces:  In the first topic of the year, I teach to write "up forces = down forces" and "left forces = right forces."  Negative signs never come up:  a force has a direction "to the left" or "up and right, 30 degrees above the horizontal," but there's no "negative" direction.

Newton's second law:  The rule I teach is to write, for example, "up forces - down forces = ma."  But sometimes a problem requires "down forces - up forces = ma."  How do my students know the difference?  I tell them always to start in the direction of acceleration.  Then all force values are plugged in without ever including a negative sign, just as I taught in the equilibrium unit.

I know many folks teach ΣF = ma, assign a positive and negative direction, then just add algebraically all forces with attention to signs.  Go for it, as numerous good physics teachers have had success with this approach.  But I personally have found that causes more confusion than it eliminates.  Understanding intuitively that "negative" in this context* simply means "to the left" is a sophisticated piece of abstract thinking.  I know that you understand, 'cause you're reading this blog and so are presumably a well-practiced physics teacher.  But consider your audience of first year students with no experience whatsoever of applying mathematical methods to the physical world.

*But not in every context, of course -- scalars can be negative, opening up a different can of worms

Sure, I could say the same thing about kinematic acceleration and momentum, two of the topics that I teach with negative signs (as I discussed yesterday).  A number of introductory-level problems in kinematics and momentum cannot be solved without including minus signs in the calculation; and so I use negatives freely and appropriately in these topics.

However, Newton's law problems never require algebraic methods with negative signs; the finessed approach detailed above will always work.

Conservation of mechanical energy:  There's no getting around the sign of mechanical work done by a force.  But try teaching that the lowest position in a problem is automatically the y=0 position for gravitational potential energy -- then no minus signs in mgh.  Elastic potential energy cannot be negative, because (1/2)kx2 will always be positive.  (No need to define x as positive or negative at all -- x is just the distance from the unstretched string length.)  Calculus-based courses must include negative signs in order to relate potential energy functions to conservative force functions, true.  Now you see one reason that I don't teach calculus-based physics as a first year course.

Spring force:  I know books write F = -kx.  In calculus-based physics, it's necessary to use the negative sign when deriving the period of a mass on a spring using a differential equation.  However, for all calculations in algebra-based introductory physics, the negative sign is an unnecessary distraction.  As with all forces, define the magnitude of the force as kx, and the direction is always toward the equilibrium point.  Done.

(If you allow students to read or use F = -kx, I will bet most of my life savings that someone, sometime in the year will state a spring constant as a negative value.  Send winnings to my school address, please.)

Electric fields and forces:  The "bible equation" defining an electric field is F = qE.  My classes practice for nearly a week: never plug in a negative sign into this equation, even if the charge experiencing the electric field is negative.  Instead, memorize the facts.  Positive charges are forced in the direction of E, negative charges are forced opposite E.  I take off mongo points if I see a negative sign included here.  My students are not sophisticated enough to use signs as indicators of direction for electric fields, then to turn around the next week to use signs properly in electric potential, where the signs have an entirely different meaning.  Why not avoid as much confusion as I can?  Signs are universally unnecessary when dealing with anything relating to a force.

Signs on electric potential are necessary because a potential could possibly be "below zero," and so a positive charge would be forced from -20 V to -40 V and not the other way.  And a negative charge will produce a "below zero" potential.

Application of above paragraph to problem solving is far more confusing than English homophone rules, and your students certainly always use there possessive pronouns correctly, eh?  So why confuse them more by including even MORE meanings of the negative sign in electrostatics?