Gravitational Potential Section 5.2

2. 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 PotentialΦ: g  -Φ • Analogous to the definition of the electrostatic potential from the electrostatic field E  -Φe

3. Gravitational PotentialΦ: 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  - Φg is a conservative field! • For a point mass:g  - [GM/r2] er (2) Φ = Φ(r) (no angular dependence!)  = (d/dr) eror Φ = (dΦ/dr) er Comparing with (2) gives: Potential of a Point Mass:Φ = -G(M/r)

4. 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 Φby 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 lineΓ

5. 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)  Thework/unit mass to move a body from position r1to position r2in a gravitational field = the potential difference between the 2 points: W= ∫dW =∫dΦΦ(r2) - Φ(r1)

6. Work/unit mass to move a body from position r1to position r2in a g field: W = ∫dW = ∫dΦΦ(r2) - Φ(r1) • Positions r2, r1 are arbitrary  Take r1   & defineΦ 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

7. 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 Φfor a point mass, Φ = -G(M/r), leadsEXACTLYto the force given by the Universal Law of Gravitation (as it should)! That is, we should get the expression: F = - [G(mM)/r2] erIntegral over volume V! • Student exercise: Show this!

8. Note: The gravitational potential Φ& gravitational potential energy (PE) of a body UINCREASEwhen work is done ON the body. • By definition, Φisalways < 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 usagesays 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  