Friday, May 8, 2015

EJSS gravity 07

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

symbolg = - G M r 2  φ = - GM r
nameField strengthPotential
unitsN kg-1 or m s-2J kg-1
meaningForce per unit massPotential energy per unit mass
quantityvectorscalar
equation|g| = G M r 2 towards the centre of the source massϕ = - G M r 
relationship to massForce, F = G M 1 M 2 r 2 = mgPotential 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 = -  ϕ  t 
g = -  ϕ  t
relationship between F and U
F = -  U  t 
F = -  U  t

Model

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