Graph of |Fg| vs. r for a solid sphere of uniform density r:

1
I.B. Gravity of a Solid Sphere

• Graph of Fg vs. r for a solid sphere of uniform
  density r
• Fg(inside) Gm(r)m/ r2 G(4p/3)rr3/r2 r.
• Fg(outside) GMEm/ r2 r-2.

Fg
R
1/r2
r
r
(I.B.1)
(I.B.2)
2
I.B. Gravity of a Solid Sphere

• 2. Graph of Fg vs. r for a solid sphere with
  M(r) Mtotr2
• Fg(inside) Gm(r)m/ r2 GMtotr2/r2
  constant.
• Fg(outside) GMEm/ r2 r-2.

r r-1.
Fg
R
1/r2
r
r
(I.B.3)
(I.B.4)
3
I.B. Gravity of a Solid Sphere

• Inside a sphere, form of gravitational force
  depends on the distribution of matter--the
  density.
• M(r) ?rdV. (I.B.5)
• 4. Outside a sphere, form of gravitational force
  depends on the inverse square law.
• 5. Far enough away from any finte matter
  distribution, Fg r-2.
• At Earths surface Fg GmEm/RE2
• This is just the weight of mass m at the surface
  W Fg GmEm/RE2 mg, where
• g GmE/RE2. (I.B.6)

4
I.C. Gravitational Potential Energy

• For uniform gravity, U mgy (e.g., near Earths
  surface)
• We need a more general expression for U based on
  Eqn. I.A.1
• As before, we define potential energy in terms of
  the work done by the force associated with the
  potential energy
• This leads to the following definition of
  gravitational potential energy
• U GmEm/r (I.C.1)

5
I.C. Gravitational Potential Energy

• 5. For distances greater than the Earths radius
• This definition of U means that U 0 when r ?
• What about a hollow sphere?

U
RE
r
RE
U increases (becomes less negative) with
increasing r
GmEm / RE
6
I.D. Escape Speed

• 1. Consider a projectile fired from a cannon
• If gravitational force is the only force that
  does work, then mechanical energy is conserved
  K1 U1 K2 U2
• For rocket to just barely escape to large values
  of r (say r ?), K2 0
• At r ?, U2 0 as well
• Therefore K1 U1 0, where K1 and U1 are
  measured at the launch point (ground)
• 1/2mv2 (GmEm/ RE) 0
• ? vesc (2GmE)/ RE1/2 (2gRE)1/2 (I.D.1)
• 11.2 km/s 24,000 mph

7
I.E. Satellite Motion

• c) From Newtons 2nd Law
  
• GmEm / r2 marad mv2 / r, so
• vc (GmE /r)1/2 vesc/v2 (I.E.1)
• Circular Orbit Speed
• In terms of the period T of the orbit

(II.E.2)
8
I.E. Satellite Motion

• Example What is the orbital period for Low
  Earth Orbit? For LEO, assume that r RE (good
  assumption since r is typically RE 300-500 km)
  and circular orbit
• T 2pRE3/2/v(GME) 90 minutes.
• note vc(LEO) 8 km/s 17,000 mph

9
I.E. Satellite Motion

• Example What is the orbital radius for a
  geosynchronus orbit?
• Ts/TLEO 24 hrs/1.5 hrs r3/2/RE 3/2
• rs/RE 162/3 6.3, or rs 6RE 40,000 km.
• vs /vc(LEO) (RE/rS)1/2 0.4, or
• vs 0.44vc(LEO) 3 km/s