Center of Mass

Balance a ruler on your fingertip. The balance point sits right in the middle — the 15 cm mark of a 30 cm ruler. Now tape a quarter to one end and try again. The balance point shifts toward the quarter.

That balance point is the center of mass — the single point where you can think of the entire mass of an object (or system) as concentrated. It’s the point that obeys Newton’s laws as if all the external forces were applied right there.

The MCAT uses this concept in problems ranging from simple balance to multi-object collisions to “where does the system actually move?” questions. Once you can calculate it (it’s just a weighted average), the rest is bookkeeping.

Definition and Calculation

For a system of discrete masses in one dimension:

For two dimensions, calculate $x_{cm}$ and $y_{cm}$ separately using the same formula with the appropriate coordinates.

Example. Two masses on a number line: $m_1 = 3$ kg at $x = 0$ m, and $m_2 = 1$ kg at $x = 4$ m.

$x_{cm} = \dfrac{3 \times 0 + 1 \times 4}{3 + 1} = \dfrac{4}{4} = 1 \text{ m}$

The center of mass is at $x = 1$ m — much closer to the 3 kg mass than to the 1 kg mass. That makes sense: the heavier mass “pulls” the center of mass toward itself.

Symmetric Objects

For a uniform, symmetric object, the center of mass is exactly at the geometric center — no calculation needed:

• Uniform rod → middle of the rod.
• Uniform sphere → center of the sphere.
• Uniform rectangular plate → intersection of the diagonals.
• Uniform ring → center of the ring (even though there’s no material there).
• Uniform donut → center of the donut hole.

Center of Mass and Motion

Newton’s second law applies to the center of mass of a system:

$F_{net,\,external} = M_{total} \cdot a_{cm}$

This has a few important consequences:

• External forces (gravity, applied pushes, friction with the outside world) determine how the center of mass accelerates.
• Internal forces (forces between parts of the same system) don’t affect the center of mass at all. The Earth pulls on you and you pull on the Earth — but those forces are internal to the Earth-you system, so they can’t move the system’s center of mass.
• If no external force acts, the center of mass moves at constant velocity (or stays at rest). Doesn’t matter what the parts do internally.

Center of Mass vs. Center of Gravity

For nearly all MCAT purposes, center of mass and center of gravity are the same point. They differ only when the gravitational field changes appreciably across the object — like an extremely tall building where $g$ at the top is measurably weaker than at the bottom. On the MCAT, treat them as identical.

Why It Matters for Equilibrium

An object is in stable equilibrium when its center of mass is as low as possible. Nudge a marble in a bowl and it rolls back to the bottom. The bowl shape forces the center of mass to rise when displaced, and gravity pulls it back down to the lowest point.

An object is in unstable equilibrium when its center of mass is as high as possible — like a marble balanced on top of an upside-down bowl. The slightest nudge sends the center of mass downhill, and the marble rolls away.

For a standing object to remain upright, the vertical line through its center of mass must fall inside the base of support (the area enclosed by the object’s contact points with the ground). Lean too far and the line passes outside the base — gravity now creates a torque that tips you over.

This is why:

• A wide stance is more stable than a narrow one (bigger base).
• People with their arms out feel more stable on a balance beam (low CoM, plus arms help keep the line over the beam).
• The Leaning Tower of Pisa is still standing — barely. Its center of mass is just over its base. If it leans much more, it falls.