AI Tutor
Try balancing a heavy metal hammer horizontally on one finger — your finger won't end up anywhere near the geometric midpoint of the handle.
Instead, it balances right next to the heavy iron head at a single point called the center of mass.

The Center of Mass

The center of mass is the unique position where the entire mass of a system can be treated as concentrated when analyzing translational motion.
Around 250 BCE, the ancient Greek mathematician Archimedes first formalized this idea while determining how unequal weights balance on a lever.

Why the Formula Works

For two masses m1m_1 and m2m_2 on a line, perfect rotational balance about the balance point xcmx_{\text{cm}} requires zero net torque: m1(xcm−x1)=m2(x2−xcm)m_1(x_{\text{cm}} - x_1) = m_2(x_2 - x_{\text{cm}}).
Rearranging this balance condition gives the mass-weighted average position: xcm=m1x1+m2x2m1+m2x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}
For any collection of NN discrete point particles with total mass M=∑miM = \sum m_i, the general 1D formula is: xcm=1M∑i=1Nmixix_{\text{cm}} = \frac{1}{M}\sum_{i=1}^{N} m_i x_i
One-dimensional balance is straightforward, but what happens when particles are scattered across a two-dimensional plane?

Multi-Dimensional Discrete Systems

Because Cartesian coordinate axes are perpendicular and independent, you calculate each spatial coordinate separately using the exact same weighting rule.