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📘 Lesson 16 of 19 · Mechanics

⚖️ Center of Mass

The center of mass is the single point where an object's entire mass can be thought of as concentrated — a powerful simplification for analyzing complex motion and balance.

Course progress: 84%

01 Key Concepts

Center of Mass Definition

The average position of all the mass in a system, weighted by how much mass is at each location.

Center of Mass Formula (Discrete System)

x_cm = (sum of m_i*x_i) / (sum of m_i), applied separately for each coordinate (x, y, z).

Center of Mass of Symmetric Objects

For a uniform, symmetric object (like a sphere, disk, or rectangle), the center of mass is exactly at its geometric center.

Motion of the Center of Mass

The center of mass of a system moves as if all external forces were applied directly to a single point mass equal to the system's total mass -- internal forces (like an explosion) don't affect the center of mass's overall path.

Stability and Center of Mass

An object is more stable when its center of mass is lower and positioned within its base of support; it tips over once the center of mass shifts outside that base.

02 Key Formulas

03 Solved Examples

Example 1 Two masses are on a line: 2 kg at x=0 and 4 kg at x=6. Find the center of mass.
  1. Apply the formula: x_cm = (2*0 + 4*6) / (2+4) = (0+24)/6.
Answer: x_cm = 4
Example 2 Three equal masses (1 kg each) are at x=0, x=3, x=6. Find the center of mass.
  1. Apply the formula: x_cm = (1*0+1*3+1*6)/(1+1+1) = 9/3.
Answer: x_cm = 3 (the middle position, as expected for equal masses evenly spaced)
Example 3 Why does a tightrope walker carry a long balancing pole?
  1. The pole adds mass far from the walker's body, lowering and stabilizing the combined center of mass.
  2. This makes it harder for the center of mass to shift outside the narrow base of support (the rope).
Answer: It lowers and stabilizes the system's center of mass, making balance easier

04 Practice Questions

1Two masses: 3 kg at x=0, 3 kg at x=8. Find center of mass.
x_cm=(0+24)/6=4
2Two masses: 1 kg at x=0, 3 kg at x=4. Find center of mass.
x_cm=(0+12)/4=3
3Where is the center of mass of a uniform sphere?
At its geometric center
4What happens to a system's center of mass motion if only internal forces act (like an internal explosion)?
It continues along its original path unaffected -- internal forces don't change the center of mass's trajectory
5What makes an object more stable in terms of its center of mass?
A lower center of mass positioned within its base of support

📄 Center of Mass — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet