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7.3.8 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 d r = - G M ( - r 2 ) , which has the same expression as g.
  3. Hence, mathematically  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 m ϕ d r Activity To do

ICT inquiry worksheet 1 (E), as well as the "G field and potential" EJS here. The HTML5 version is below, the Java worksheet customization to HTML5 is work in progress. 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 Gravitational force per unit mass Gravitational potential energy per unit mass
quantity vector scalar
equation |g| = G M r 2 towards the centre of the source mass
ϕ = - GM r
relationship to mass Force, F = G M 1 M 2 r 2 = mg Potential energy, U = -m G M r = mϕ
graph model g = -6.67*500/(abs(r)*r)
model phi = -6.67*500/abs(r)
computer model if M = 500.  -6.67*500/(abs(r)*r) -6.67*500/abs(r)
relationship between g and ϕ
relationship between g and φ
g = - d ϕ d r
relationship between F and U
relationship between F and U
F = - d U d r Model

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This email address is being protected from spambots. You need JavaScript enabled to view it.; Anne Cox; Wolfgang Christian; Francisco Esquembre

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