# 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  r = - G M - r 2 , which has the same expression as g.
3. Hence, mathematically $\genfrac{}{}{0.1ex}{}{d}{}$ ϕ  r = G M r 2 = - g
4. To understand the meaning of $g$ = -  ϕ  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, $g$ = -  ϕ  r

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

$m$ g = -  m ϕ  r
thus $F$ = -  U  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

https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_gravity07/gravity07_Simulation.xhtml