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← Unit 2 Handouts

Topic 2.1: Systems and Center of Mass, Part 2 — Objects on a Plane

Name: _______________________________ Date: _______________

Before We Start: Yesterday's Recap

Yesterday we found the center of mass of objects lined up along ONE line. Without looking back: what's the formula, and what's the #1 mistake to avoid?

Today's New Concepts

AP CED Alignment: Unit 2, Topic 2.1 (continued)

Center of Mass in Two Dimensions

  • Objects scattered across a plane need TWO coordinates: $x_{cm}$ and $y_{cm}$
  • Same formula, applied separately to each direction:

$$x_{cm} = \dfrac{\sum m_ix_i}{\sum m_i} \qquad\qquad y_{cm} = \dfrac{\sum m_iy_i}{\sum m_i}$$

  • Nothing new mathematically — just do the weighted average twice, once for x's, once for y's

Symmetry in Two Dimensions

  • If a 2D object has TWO lines of symmetry, the center of mass is at the intersection of those two lines
  • A system arranged with point symmetry (every object has an identical mirror partner across one central point) has its center of mass exactly at that central point

Predicting Shifts — Direction and Bounds

  • A system's center of mass always lies between the centers of mass of its individual parts — never outside that range
  • If one part doesn't move, the system's center of mass can only shift by an amount smaller than however far the other part moved

Worked Example — 2D Calculation:

  • Satellite A: 2 kg at (0, 0). Satellite B: 3 kg at (4, 0). Satellite C: 5 kg at (2, 6).
  • $x_{cm} = \dfrac{(2)(0)+(3)(4)+(5)(2)}{10} = \dfrac{22}{10} = 2.2\text{ m}$
  • $y_{cm} = \dfrac{(2)(0)+(3)(0)+(5)(6)}{10} = \dfrac{30}{10} = 3.0\text{ m}$
  • Center of mass: (2.2 m, 3.0 m)

Worked Example — Direction and Bounds:

  • A board's own center of mass is fixed at 50 cm. A block starts at 20 cm and slides to 50 cm.
  • The system's center of mass shifts toward 50 cm (same direction the block moved), but by less than 30 cm — since the board's own center never moved, it anchors part of the average in place.

Keep This Sheet!

These sheets build into your semester study guide. Keep them in order in a binder or folder — you'll want to flip back through them before quizzes, unit tests, and when AP exam review starts in the spring.