Resource Lesson
Universal Gravitation and Weight
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The quantity of matter present in an object is known as the object's
mass
. It is used to measure an object's inertia and does not depend on the object's location. Often an object's mass is calculated indirectly by using the relationship
D = M/V
or
density = mass/volume
. If the object's volume and the density of the substance comprising the object are known, then the object's mass can be determined as the product of density x volume. Other specific methods are available to determine an object's
inertial or gravitational mass
depending on available equipment.
A concept directly related to the scalar quantity mass is the vector quantity
weight
. On earth, an object's weight is represented by a vector whose direction points directly towards the earth's center of mass and whose magnitude depends on the object's location in the earth's gravitational field. Weight is calculated as the product of the object's mass (in kg) and the local gravitational field strength (in N/kg) which is also known as the acceleration due to gravity (in m/sec
^{2}
). The equation is
wt = mg
.
To derive an expression for the local gravitational field strength, or acceleration due to gravity, we will start with a statement of
Newton's Law of Universal Gravitation
(1686).
Every object in the universe attracts every other object in the universe with a force whose magnitude is directly proportional to the product of their masses and is inversely proportional to the square of the distance between their centers.
This law can be stated mathematically as
where G is a very small constant that was first experimentally determined by
Henri Cavendish in 1798
. Today it's accepted value is 6.672 x 10
^{-11}
Nm
^{2}
/kg
^{2}
. Since G is so very small, the gravitational attraction between small masses on the earth's surface is not noticeable compared with the gravitational attraction between either mass and the earth. Gravitation is only an attractive force unlike electromagnetic forces which are both attractive and repulsive. Moreover, since the majority of the universe is neutral, it is gravity that holds it together.
The region surrounding a massive object in which another object having mass would feel a gravitational force of attraction is called a
gravitational field
.
To calculate the strength of this field, we will start with two expressions for an object's weight
where
m
represents the mass of the first object being placed in the field of a second larger mass,
M
, and
r
represents the distance between their centers. The "m's" cancel since they are the same on both sides of the equation, giving us the following expression for the gravitational field's strength at a distance
r
from the center of mass,
M
.
Gravitational fields are an example of
radial, inverse-square fields
. Shown below is a diagram of how the strength of a gravitational field varies from the center of a solid, uniform sphere.
On the surface of the earth we can determine that
g
has a numerical value of 9.81 by using the following accepted values:
G = 6.67 x 10
^{-11}
Nm
^{2}
/kg
^{2}
M
_{earth}
= 5.98 x 10
^{24}
kg
R
_{earth}
= 6.37 x 10
^{6}
m
The units on
g
can be expressed as either N/kg or m/sec
^{2}
. When expressed as 9.81 m/sec
^{2}
,
g
is known as the
acceleration due to gravity
. When expressed as 9.81 N/kg,
g
is referred to as the earth's
gravitational field strength
at its surface.
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