Center of mass calculations for AP Physics 1

The revisions to the AP Physics exams for 2025 are imminent.  I've posted about the formatting changes, which are truly no big deal for physics 1/2, and are so, so welcome in physics C.  (Doubling the time available for the exam, but not doubling the length of the exam, will allow students the time they need to approach complicated questions.)

The *content* changes are nonexistent in physics C. In P1, most folks are focused on the major change that adds fluid mechanics.  But what about the minor changes to P1?  They're there, too: the parallel axis theorem, quantitative questions about gravitational potential energy in orbits, and calculations with center of mass.

So what is there to know about center of mass quantitatively in AP Physics 1?

First of all, the conceptual treatment of center of mass motion that's already covered on this exam won't go away.  We still need to understand that the center of mass of a system obeys Newton's laws - with no unbalanced external force, the center of mass moves in a straight line at constant speed; with an unbalanced external force F, the acceleration of the center of mass is F/M where M is the system mass.

Then, the conceptual understanding of the location of a system's center of mass still is relevant.  For a symmetric object, the center of mass is in the, um, center.  For two equal-mass objects, the center of mass is right in between.  And for two unequal-mass objects, the center of mass is closer to the larger-mass object.

The new stuff is based on calculating the position of the center of mass using the equation m1x1 + m2x2 = Mtotal(xcm).  No, please don't write this equation using the notation from the updated official equation sheet, with summation notation!  Your students don't know what that ziggy-zaggy capital E is; and why is there an i in there?  We have to know about imaginary numbers now?!, they'll ask.*

*Yes, I'm aware that students must be able to calculate for more than two objects, in which case the equation I wrote is technically incomplete.  Students will figure out how to add a third or fourth object just fine once they have experience calculating for two objects.  In first year physics, complicated but precise mathematical notation obscures rather than elucidates meaning.  Sort of like any statement that includes the word "technically."

The AP Physics 1 exam famously has minimal use for numerical answers.  Only one of the ten revised "science practices", with which each AP Physics 1 exam question must align, includes calculating numerical quantities.  So while "here are two objects and their positions, calculate the location of their center of mass" is a legitimate question, expect this end-of-textbook-chapter-style problem to be rare.  But what else is there?  Think of the same kinds of questions that are asked about kinematics or energy:

* How would the center of mass position change if the left-hand object's mass were doubled?  If both masses were doubled?  If a third object were placed somewhere?

* Graph the position of the center of mass as a function of the mass of one object; graph the position of the center of mass as a function of time with this known external force acting on the system.

* Describe an experiment which will determine the center of mass position for this set of objects on a thin plank.  Now here's some data from that experiment; plot the data and use the slope of a best-fit line to determine the center of mass position.

All these questions start with the center of mass formula, but use the formula to make predictions, graphs, experimental conclusions.  And that's AP Physics 1 in a nutshell.