12 ORBITAL MOTION 12.4 Gravitational potential energy∞∫∞⊕Body
Sun
Moon
Mercury
Venus
Earth
Mars
Jupiter
SaturnM/M⊕
3.33 × 105
0.0123
0.0553
0.816
1.000
0.108
318.3
95.14R/R⊕ g/g⊕
109.0 28.1
0.273 0.17
0.383 0.38
0.949 0.91
1.000 1.000
0.533 0.38
11.21 2.5
9.45 1.07(^)
Table 6: The mass, M, radius, R, and surface gravity, g, of various bodies in the Solar System. All
quantities are expressed as fractions of the corresponding terrestrial quantity.
is the change in the potential energy of the first mass associated with this shift?
According to Eq. (5.33),
U( ) − U(r) = − [−f(r)] dr. (12.10)
r
There is a minus sign in front of f because this force is oppositely directed to the
motion. The above expression can be integrated to give
G M m
U(r) = −.^ (12.11)^
r
Here, we have adopted the convenient normalization that the potential energy
at infinity is zero. According to the above formula, the gravitational potential
energy of a mass m located a distance r from a mass M is simply −G M m/r.
Consider an object of mass m moving close to the Earth’s surface. The potential
energy of such an object can be written
U = −
G M⊕ m
, (12.12)
R⊕ + z
where M⊕ and R⊕ are the mass and radius of the Earth, respectively, and z is the
vertical height of the object above the Earth’s surface. In the limit that z R⊕,
the above expression can be expanded using the binomial theorem to give
U ' −
G M⊕ m
- G M⊕ m
z, (12.13)
R R 2
⊕