Equivalence Principle

Inertial Mass and Gravitational Mass

Every object has two distinct properties involving mass:

Inertial mass $m_i$ quantifies an object's resistance to changes in motion. When a net force $\vec{F}_{\text{net}}$ acts on an object, Newton's second law relates force to acceleration:

$$\vec{F}_{\text{net}} = m_i \vec{a}$$

An object with larger inertial mass experiences less acceleration for the same applied force.

Gravitational mass $m_g$ determines the strength of gravitational attraction between objects. The gravitational force between two objects with gravitational masses $m_{g1}$ and $m_{g2}$ separated by distance $r$ is:

$$F_g = G \frac{m_{g1} m_{g2}}{r^2}$$

where $G$ is the gravitational constant. Near Earth's surface, an object with gravitational mass $m_g$ experiences gravitational force:

$$F_g = m_g g$$

where $g$ is the gravitational field strength.

Remarkably, inertial mass and gravitational mass are experimentally verified to be equivalent. Every measurement confirms $m_i = m_g$ to extraordinary precision, which allows us to use a single symbol $m$ for both. This equivalence is not obvious—these masses arise from completely different physical contexts.

The Equivalence Principle

The equivalence principle states that an observer in a noninertial reference frame cannot distinguish between an object's apparent weight and the gravitational force exerted on the object by a gravitational field.

Consider an observer in an upward-accelerating elevator with acceleration $a$. An object of mass $m$ inside experiences two forces in the inertial (ground) frame: the normal force $N$ upward and gravitational force $m g$ downward. Applying Newton's second law in the vertical direction:

$$N - m g = m a$$ $$N = m(g + a)$$

...