Acceleration Due to Gravity

Physical Law

A body $B$ in a uniform gravitational field experiences a force which gives rise to a constant acceleration independent of the mass of the body.

If the force due to the gravitational field is the only force on $B$, it is said to be in free fall.

Derivation

This law can be derived from Newton's Law of Universal Gravitation.

Let the mass of $B$ be $m$.

Let the mass of the body $P$ which gives rise to the gravitational field be $M$.

Then the force on $B$ is given by:

$F = G \dfrac {M m} {r^2}$

where:

$G$ is the universal gravitational constant

$r$ is the distance between the centers of gravity of $B$ and $P$.

The assumption is that $M$ is orders of magnitude greater than $m$, and $r$ is also several orders of magnitude greater than the displacements observed on $B$ in the local frame.

Then:

$F = m \dfrac {G M} {r^2}$

But from Newton's Second Law of Motion:

$F = m a$

where $a$ is the magnitude of the acceleration which would be imparted to $B$ if no other force were acting on it.

Hence:

$a = \dfrac {G M} {r^2}$

which is, as we said, independent of the mass of $B$.

$\blacksquare$

Gravity on Earth

When the body $P$ of mass $M$ is the Earth, and the body $B$ of mass $m$ is located at or near its surface, it is usual to use $g$ for the local gravitational constant $\dfrac {G M} {r^2}$.

Therefore the force on $B$ can be expressed as:

$F = m g$

Historical Note

This result famously contradicts Aristotle, who taught that heavier objects fall faster than light ones.

There exists a well-known story about how Galileo proved this law by dropping two objects of different weights from a height (supposedly the Leaning Tower of Pisa).

Apparently this never actually happened.

However, Simon Stevin did perform a similar experiment some time before Galileo started experimenting.

The fact that in the general case air resistance can not be ignored goes some way to explaining how the truth was not arrived at earlier.

Even children notice how leaves, for example, flutter gently down from on high, whereas stones tend to plummet.

Sources

• 1966: Isaac Asimov: Understanding Physics ... (previous) ... (next): $\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Free Fall