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Mechanics · Quick Reference
Center of Mass Cheat Sheet
In one line: 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.
Key Ideas
1Center of Mass Definition. The average position of all the mass in a system, weighted by how much mass is at each location.
2Center 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).
3Center 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.
4Motion 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.
5Stability 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.
Worked Examples
Two masses are on a line: 2 kg at x=0 and 4 kg at x=6. Find the center of mass.
x_cm = 4
Three equal masses (1 kg each) are at x=0, x=3, x=6. Find the center of mass.
x_cm = 3 (the middle position, as expected for equal masses evenly spaced)
Why does a tightrope walker carry a long balancing pole?
It lowers and stabilizes the system's center of mass, making balance easier
Formulas
x_cm = (sum of m_i * x_i) / (sum of m_i)
Practice Yourself
Two masses: 3 kg at x=0, 3 kg at x=8. Find center of mass.
x_cm=(0+24)/6=4
Two masses: 1 kg at x=0, 3 kg at x=4. Find center of mass.
x_cm=(0+12)/4=3
Where is the center of mass of a uniform sphere?
At its geometric center