The Value of g
In the previous topic, an equation was
given for determining the force of gravity (Fgrav) with which an
object of mass m was attracted to the earth
Fgrav = m*g
Now in this unit, a second equation has been introduced for
calculating the force of gravity with which an object is attracted to the earth.
where d represents
the distance from the center of the object to the center of the earth.
In the first equation above, g is
referred to as the acceleration of gravity. Its value is 9.8 m/s2 on Earth.
That is to say, the acceleration of gravity on the surface of the earth at sea
level is 9.8 m/s2. When discussing the
acceleration of gravity, it was mentioned that the value of g is dependent upon
location. There are slight variations in the value of g about earth's surface.
These variations result from the varying density of the geologic structures
below each specific surface location. They also result from the fact that the
earth is not truly spherical; the earth's surface is further from its center at
the equator than it is at the poles. This would result in larger g values at
the poles. As one proceeds further from earth's surface - say into a location
of orbit about the earth - the value of g changes still.
The
Value of g Depends on Location
To understand why the value of g is so location dependent, we
will use the two equations above to derive an equation for the value of g.
First, both expressions for the force of gravity are set equal to each other.
Now observe that the mass of the object - m - is
present on both sides of the equal sign. Thus, m can be canceled from
the equation. This leaves us with an equation for the acceleration of gravity.
The above equation demonstrates that the acceleration of
gravity is dependent upon the mass of the earth (approx. 5.98x1024 kg) and
the distance (d) that an object
is from the center of the earth. If the value 6.38x106 m (a
typical earth radius value) is used for the distance from Earth's center, then g
will be calculated to be 9.8 m/s2. And of course, the value of g
will change as an object is moved further from Earth's center. For
instance, if an object were moved to a location that is two earth-radii from
the center of the earth - that is, two times 6.38x106 m - then a
significantly different value of g will be found. As shown below, at twice the
distance from the center of the earth, the value of g becomes 2.45 m/s2.
The table below shows the value of g at various locations
from Earth's center.
|
Location |
Distance
from Earth's center |
Value of g |
|
Earth's surface |
6.38 x 106 m |
9.8 |
|
1000 km above surface |
7.38 x 106 m |
7.33 |
|
2000 km above surface |
8.38 x 106 m |
5.68 |
|
3000 km above surface |
9.38 x 106 m |
4.53 |
|
4000 km above surface |
1.04 x 107 m |
3.70 |
|
5000 km above surface |
1.14 x 107 m |
3.08 |
|
6000 km above surface |
1.24 x 107 m |
2.60 |
|
7000 km above surface |
1.34 x 107 m |
2.23 |
|
8000 km above surface |
1.44 x 107 m |
1.93 |
|
9000 km above surface |
1.54 x 107 m |
1.69 |
|
10000 km above surface |
1.64 x 107 m |
1.49 |
|
50000 km above surface |
5.64 x 107 m |
0.13 |
As is evident from both the equation and the table above, the value of g varies
inversely with the distance from the center of the earth. In fact, the variation in g
with distance follows an inverse square law where g is
inversely proportional to the distance from earth's center. This
inverse square relationship means that as the distance is doubled, the value of
g decreases by a factor of 4. As the distance is tripled, the value of g
decreases by a factor of 9. And so on. This inverse square relationship is
depicted in the graphic at the right.
Calculating
g on Other Planets
The same equation used to determine the value of g on Earth'
surface can also be used to determine the acceleration of gravity on the
surface of other planets. The value of g on any other planet can be calculated
from the mass of the planet and the radius of the planet. The equation takes
the following form:
Using this equation, the following acceleration of gravity
values can be calculated for the various planets.
|
Planet |
Radius (m) |
Mass (kg) |
g (m/s2) |
|
Mercury |
2.43 x 106 |
3.2 x 1023 |
3.61 |
|
Venus |
6.073 x 106 |
4.88 x1024 |
8.83 |
|
Mars |
3.38 x 106 |
6.42 x 1023 |
3.75 |
|
Jupiter |
6.98 x 107 |
1.901 x 1027 |
26.0 |
|
Saturn |
5.82 x 107 |
5.68 x 1026 |
11.2 |
|
Uranus |
2.35 x 107 |
8.68 x 1025 |
10.5 |
|
Neptune |
2.27 x 107 |
1.03 x 1026 |
13.3 |
|
Pluto |
1.15 x 106 |
1.2 x 1022 |
0.61 |
The acceleration of gravity of an object is a measurable
quantity. Yet emerging from Newton's universal law of gravitation is a
prediction that states that its value is dependent upon the mass of the Earth
and the distance the object is from the Earth's center. The value of g is independent of the mass of
the object and only dependent upon location - the
planet the object is on and the distance from the center of
that planet