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   ϕ  r = - G M - r 2 , which has the same expression as g.
  3. Hence, mathematically  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


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 |g| = G M r 2 towards the centre of the source mass
ϕ = - G M r 
relationship to mass Force, F = G M 1 M 2 r 2 = mg Potential energy, U = -m G M r = mφ
computer model if M = 500.  -6.67*500/(abs(r)*r) -6.67*500/abs(r)
relationship between g and φ
g = -  ϕ  t 
g = -  ϕ  t
relationship between F and U
F = -  U  t 
F = -  U  t