Introduction - An Introduction to Dynamic Meteorology - page 7

so that the total pressure gradient force per unit mass is

F
m =−

1
ρ ∇

p (1.1)

It is important to note that this force is proportional to the gradient of the pressure
field, not to the pressure itself.

1.4.2 Gravitational Force

Newton's law of universal gravitation states that any two elements of mass in the
universe attract each other with a force proportional to their masses and inversely
proportional to the square of the distance separating them. Thus, if two mass
elements M and m are separated by a distance r

(with the vector r directed
toward m as shown in Fig. 1.2), then the force exerted by mass M on mass m due
to gravitation is

≡| r |

r
r

GMm
r 2

F g =− (1.2)

where G is a universal constant called the gravitational constant. The law of grav-
itation as expressed in (1.2) actually applies only to hypothetical “point” masses
since for objects of finite extent r will vary from one part of the object to another.
However, for finite bodies, (1.2) may still be applied if

is interpreted as the
distance between the centers of mass of the bodies. Thus, if the earth is designated
as mass M and m is a mass element of the atmosphere, then the force per unit mass
exerted on the atmosphere by the gravitational attraction of the earth is

| r |

r
r

F g
m ≡

GM
r 2

g ∗ =− (1.3)

Fig. 1.2 Two spherical masses whose centers are separated by a distance r.