# acceleration due to gravity (*g*)

Figure 1. An object that is allowed to fall freely will, if
the effects of air resistance are ignored, gather speed (accelerate)
at a rate of about 9.8 m/s^{2} (32 ft/s^{2}). If dropped
from rest, it will have fallen 4.9 meters (16 feet) and be traveling at a
speed of 9.8 m/s (32 ft/s) after 1 second. After 2 seconds, it will
have fallen a further 14.7 meters (48 feet) and be traveling at 19.6 m/s
(64 ft/s). After 3 seconds, it will have fallen a further 24.5 meters (80
feet) and be traveling at 29.4 m/s (96 ft/s).

Figure 2. Leaning Tower of Pisa.

The acceleration due to gravity is the acceleration that an
object experiences because of gravity when
it falls freely close to the surface of a massive body, such as a planet.
Also known as the **acceleration of free fall**, its value
can be calculated from the formula

*g*=

*GM*/ (

*R*+

*h*)

^{ 2}

where *M* is the mass of the gravitating
body (such as the Earth), *R* is the radius of the body, *h* is
the height above the surface, and *G* is the gravitational
constant (= 6.6742 × 10^{-11} N·m^{2}/kg^{2}).
If the falling object is at, or very nearly at, the surface of the gravitating
body, then the above equation reduces to

*g*=

*GM*/

*R*

^{ 2}

In the case of the Earth, *g* comes out to be approximately 9.8 m/s^{2} (32 ft/s^{2}), though the exact value depends on location because
of two main factors: the Earth's rotation and the Earth's equatorial
bulge.

## Why g varies from place to place

The downward force of gravity is opposed by an outward centrifugal
force due to the planet's rotation, which is greater at the equator
than at a higher latitudes. (The centrifugal force is "fictitious" in the
sense that the real force caused by rotation is the centripetal
force; however, it is a convenient fiction for the sake of calculations.)
By itself, this effect would result in a range of values of *g* from
9.789 m/s^{2} (32.116 ft/s^{2}) at the equator to 9.823
m/s^{2} (32.228 ft/s^{2}) at the poles. This discrepancy
is further accentuated because of the Earth's equatorial bulge, which causes
objects at lower latitudes to be further from the planet's center than objects
nearer the poles and hence subject to a slightly weaker gravitational pull.

Overall these two effects result in a variation of 0.052 m/s^{2} (0.171 ft/s^{2}) in the value of *g*, which leads to a variation
in the weight of an object by about 0.5%
depending on whether it is weighed at the equator or at one of the poles.
Taking an average over the whole surface of the Earth, physicists have arrived
at a standard value for *g* of 9.80665 m/s^{2} (32.1740 ft/s^{2}).
On other planets and moons the values of the acceleration due to gravity
may be very different, resulting in different weights for the same object
on these various worlds.

## The effect of air resistance

If the Earth had no atmosphere, an object dropped from a great height would
keep accelerating at a rate of 9.8 m/s^{2} (32 ft/s^{2})
until it hit the ground. For example, if a person fell from an aircraft
at an altitude of 10,000 meters (32,808 feet), they would be traveling at about
442 m/s (1,450 ft/s) by the time they landed. In practice, this doesn't
happen because of air resistance. The faster an object falls, the greater
is the air resistance acting on it. At a certain velocity, known as the **terminal velocity**, the downward force of gravity is balanced
out by the upward force of air resistance and there is no further acceleration.

If there were no atmosphere, all objects would fall at the same rate. This
happens, for example, on the Moon. In one of
the most memorable moments of the space program, David Scott,
commander of the Apollo 15 mission, standing
on the Moon's surface, dropped two objects – a geological hammer and
a falcon's feather (the Apollo 15 lunar module was called *Falcon*)
– at the same time from the same height. The feather didn't drift
down, meanderingly, as it would have done on Earth. Instead, in the airless
vacuum of space, it fell straight, without a flutter, keeping pace with
the hammer and reaching the lunar surface at the same instant.

## Galileo and the Leaning Tower

One of the most famous stories in science is about Galileo Galilei and the Leaning Tower of Pisa. Galileo supposedly reached out from an upper balcony and let fall two stones of different weights. A remarkable thing happened: to the gasps and amazement of the crowd below, the stones hit the ground together. Although doubtless in part apocryphal, the account does at least have some backing from Galileo's pupil and amanuensis Viviani, who reported that Galileo had done the experiment "in front of all the faculty and students assembled."

## g-force

Units of *g*, or "gee," are used in aerospace when describing loads
on aircraft or spacecraft. Someone who carried out remarkable experiments
(on himself) involving very high g-forces was the pioneering aerospace physician
John Stapp.

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