Relationship between F and U; between g and φ

To understand how g is related to φ:

1. Similarly, compare $$ g = - G M r 2 and $$ φ = - GM r in the above table.
2. If we differentiate $$ φ = - GM r    with respect to r, we will get  $\genfrac{}{}{0.1ex}{}{d}{}$ ϕ ⅆ r = - G M - r 2 , which has the same expression as g.
3. Hence, mathematically $\genfrac{}{}{0.1ex}{}{d}{}$ ϕ ⅆ r = G M r 2 = - g
4. To understand the meaning of $g$ = - ⅆ ϕ ⅆ r observe the two graphs carefully, on the right side where r is positive, the gradient of φ vs r graph is positive but the value of g will be negative. And on the left side where r is negative, the gradient of φ vs r graph is negative but the value of g is positive. Thus, $g$ = - ⅆ ϕ ⅆ r

Similarly, it can be concluded that by multiplying both sides by test mass m.

$m$ g = - ⅆ m ϕ ⅆ r 
thus $F$ = - ⅆ U ⅆ r

Summary

symbol	$g$ = - G M r 2	$$ φ = - GM r
name	Field strength	Potential
units	N kg-1 or m s-2	J kg-1
meaning	Force per unit mass	Potential energy per unit mass
quantity	vector	scalar
equation	$\mathrm{|g|}$ = G M r 2 towards the centre of the source mass	$\varphi$ = - G M r
relationship to mass	Force, $F$ = G M 1 M 2 r 2 = mg	Potential energy, $U$ = -m G M r = mφ
graph
computer model if M = 500.	-6.67*500/(abs(r)*r)	-6.67*500/abs(r)
relationship between g and φ	$g$ = - ⅆ ϕ ⅆ t  $g$ = - ⅆ ϕ ⅆ t
relationship between F and U	$F$ = - ⅆ U ⅆ t  $F$ = - ⅆ U ⅆ t

Model