Gravitational Potential Section 5.2 Start with the Gravitational Field –Point mass: g  - [GM/r 2 ] e r –Extended body: g  - G ∫ [ρ(r)dv/r 2 ]e r 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  -  Φ e

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/r 2 )   g = 0  g  -  Φ g is a conservative field! For a point mass: g  - [GM/r 2 ] e r (2)  Φ = Φ(r) (no angular dependence!)   = (d/dr) e r or  Φ = (dΦ/dr) e r Comparing with (2) gives: 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 = ∫ρ s (r)da) Φ = -G ∫[ρ s (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 = -gdr  (  Φ)  dr =  i (  Φ/  x i )dx i  dΦ This is true because Φ is a function only of the coordinates of the point at which it is measured: Φ = Φ(x 1,x 2,x 3 )  The work/unit mass to move a body from position r 1 to position r 2 in a gravitational field = the potential difference between the 2 points: W= ∫dW = ∫dΦ  Φ(r 2 ) - Φ(r 1 )

Work/unit mass to move a body from position r 1 to position r 2 in a g field: W = ∫dW = ∫dΦ  Φ(r 2 ) - Φ(r 1 ) Positions r 2, r 1 are arbitrary  Take r 1   & 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)/r 2 ] e r 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  