This force is derivable from the potential function
φ=
k
ρ, where ρ=
√
(x−x 1 )^2 + (y−y 1 )^2 + (z−z 1 )^2
is the distance between the two masses and k=Gm 1 m 2 is a constant. In the above
relation the coordinates of the center of mass of the bodies 1 and 2 are respectively
P 1 (x 1 , y 1 , z 1 )and P 2 (x, y, z ). The quantity kis a constant, and eˆρis a unit vector with
origin at P 1 and pointing toward P 2 .The force of attraction of mass m 1 toward mass
m 2 is calculated by the vector operation F =−grad φ. To calculate this force, first
calculate the partial derivatives
∂φ
∂x =
∂φ
∂ρ
∂ρ
∂x =
−k
ρ^2
(x−x 1 )
ρ
∂φ
∂y =
∂φ
∂ρ
∂ρ
∂y =
−k
ρ^2
(y−y 1 )
ρ
∂φ
∂z
=∂φ
∂ρ
∂ρ
∂z
=−k
ρ^2
(z−z 1 )
ρ
and then the gradient is calculated and one obtains
F =−grad φ= k
ρ^2
[
(x−x 1 )
ρ
ˆe 1 +(y−y^1 )
ρ
ˆe 2 +(z−z^1 )
ρ
ˆe 3
]
=
k
ρ^2
ˆeρ
Here r =xˆe 1 +yˆe 2 +zˆe 3 is a position vector for the point P 2 and r 1 =x 1 ˆe 1 +y 1 ˆe 2 +z 1 ˆe 3
is a position vector for the point P 1. The vector r −r 1 is a vector pointing from P 1 to
P 2 and the vector
r −r 1
|r −r 1 |=
ˆeρis a unit vector pointing from P 1 to P 2 .Here the vector
field is called conservative since the force field is derivable from a potential function.
The potential function for Newton’s law of gravitation is called the gravitational
potential. By using the relation F = +grad φone obtains the force of attraction of
mass m 2 toward mass m 1.
