Download Sect. 5.2 (IA)

Gravitational Potential
Section 5.2
• Start with the Gravitational Field
– Point mass: g  - [GM/r2] er
– Extended body: g  - G ∫[ρ(r)dv/r2]er
Integral over volume V
• These should remind you of expressions for the electric
field (E) due to a point charge & due to an extended
charge distribution. Identical math, different physics!
• Define: Gravitational Potential Φ: g  -Φ
– Analogous to the definition of the electrostatic
potential from the electrostatic field E  -Φe
Gravitational Potential Φ: g  -Φ (1)
Dimensions of Φ : (force/unit mass)  (distance) or
energy/unit mass. The mathematical form, (1), is justified by:
g (1/r2)   g = 0  g  - Φ
g is a conservative field!
• For a point mass:
g  - [GM/r2] er (2)
 Φ = Φ(r) (no angular dependence!)
  = (d/dr) er or Φ = (dΦ/dr) er
Comparing with (2) gives:
Potential of a Point Mass: Φ = -G(M/r)
Potential of a Point Mass: Φ = -G(M/r)
• Note: The constant of integration has been ignored! The
potential Φ is defined only to within additive constant.
Differences in potentials are meaningful, not absolute Φ .
Usually, we choose the 0 of Φ by requiring Φ 0 as r  
• Volume Distribution of mass (M = ∫ρ(r)dv):
Φ = -G ∫[ρ(r)dv/r]
Integral over volume V
Surface Distribution: (thin shell; M = ∫ρs(r)da)
Φ = -G ∫[ρs(r)da/r] Integral over surface S
Line Distribution: (one d; M = ∫ρ(r)ds)
Φ = - G ∫[ρ(r)ds/r]
Integral over line Γ
Physical significance of the gravitational potential Φ?
– It is the [work/unit mass (dW) which must be done by an
outside agent on a body in a gravitational field to displace
it a distance dr] = [force  displacement]:
dW = -g•dr  (Φ)dr = i(Φ/xi)dxi  dΦ
This is true because Φ is a function only of the coordinates
of the point at which it is measured: Φ = Φ(x1,x2,x3)
 The work/unit mass to move a body from
position r1 to position r2 in a gravitational field =
the potential difference between the 2 points:
W= ∫dW = ∫dΦ  Φ(r2) - Φ(r1)
• Work/unit mass to move a body from position r1 to
position r2 in a g field:
W = ∫dW = ∫dΦ  Φ(r2) - Φ(r1)
• Positions r2, r1 are arbitrary  Take r1   &
define Φ  0 at   Interpret Φ(r) as the work/unit
mass needed to bring a body in from  to r.
• For a point mass m in a gravitational field
with a potential Φ, define:
Gravitational Potential Energy: U  mΦ
Potential Energy
• For a point mass m in a gravitational potential Φ
Gravitational Potential Energy: U  mΦ
• As usual, the force is the negative gradient of the
potential energy  the force on m is F  - U
– Of course, using the expression for Φ for a point
mass, Φ = -G(M/r), leads EXACTLY to the force
given by the Universal Law of Gravitation (as it
should)! That is, we should get the expression:
F = - [G(mM)/r2] er
Integral over volume V!
– Student exercise: Show this!
• Note: The gravitational potential Φ & gravitational
potential energy (PE) of a body U INCREASE when
work is done ON the body.
– By definition, Φ is always < 0 & it  its max value
(0) as r  
– Semantics & a bit of philosophy!
A potential energy (PE) exists when a body is in a g
field (which must be produced by a source mass!).
THIS PE IS IN THE FIELD. However, customary
usage says it is the “PE of the body”.
– We may also consider the source mass to have an intrinsic
PE = gravitational energy released when body was formed
or = the energy needed to disperse the mass to r  