Inertial versus Gravitational Mass

Mass shows up in Newton's second law twice, in two different jobs. In F = ma, mass tells you how stubborn an object is: how much it resists being accelerated by any force at all, a shove, a spring, a rocket engine. Call that inertial mass. But mass also shows up in the force of gravity itself, F_grav = mg: it tells you how strongly Earth pulls on the object. Call that gravitational mass. Nothing in logic forces these two numbers to match. An object could easily resist acceleration a lot while gravity barely notices it, or the reverse. Every experiment ever done says otherwise: the two masses are identical, to extraordinary precision.

That coincidence is why free fall is so clean. Set gravitational force equal to ma: mg = ma, where the left m is gravitational mass and the right m is inertial mass. Because they are the same number, the m's cancel, leaving a = g. Mass drops out completely, so a boulder and a pebble fall at the same rate once air resistance is removed. If the two masses were different, heavier objects would fall faster or slower than light ones, and free fall would depend on composition.

You can measure the two kinds separately. Inertial mass comes from a spring: attach the object, pull it, measure the acceleration for a known force, and read mass off F = ma; this works even with no gravity around. Gravitational mass comes from a balance: compare the gravitational pull on the object to the pull on known masses. Different setups, same number, every time, which is why this topic sits right behind weight and g in the unit.