Environmental Engineering Reference
In-Depth Information
of energy since we know the total energy of the orbit and, from Example 9.5.1, that
the velocity is tangential to the orbit at the extrema of the orbit. Thus
GM
2
a
1
2
u
2
GM
r
−
=
−
,
where r is 1861 km at the apocentre or 1838 km at the pericentre. Re-arranging
allows us to determine that
2
a
r
−
1
.
GM
a
u
2
=
Substituting for a
=
1849
.
5
km, the maximum and minimum speeds are 1.64 km/s
and 1.62 km/s.
PROBLEMS 9
9.1 Prove that the gravitational potential inside a hollow spherical shell is a con-
stant.
9.2 Suppose a hole were drilled straight through the Earth along a diameter. Show
that a body dropped into the hole would execute simple harmonic motion.
9.3 The gravitational self-energy of an object is the total work done against grav-
ity in order to assemble its constituent parts from infinity. Show that the
self-energy of a uniform sphere of mass
M
is
GM
2
R
3
5
E
=−
.
If the sphere rotates about a diameter such that the sum of its rotational and
gravitational energies is zero what is its angular velocity?
9.4 A comet orbits the Sun such that the perihelion distance is 7
.
48
10
10
×
mand
10
4
ms
−
1
. (i) Determine the magnitude of
the angular momentum of the comet divided by its mass; (ii) Determine the
kinetic energy and gravitational potential energy of the comet (both divided
by the comet's mass) at the point of closest approach. Is the orbit bound or
unbound? (iii) Using the conservation of energy, in conjunction with fact that
the angular momentum is always proportional to the tangential component of
the comet's velocity, determine the radial and tangential components of the
velocity of the comet, and hence its speed, when it is a distance 1
.
50
its speed at perihelion is 5
.
96
×
10
11
×
m
from the Sun.
9.5 A star of mass
M
is located at the centre of a spherical dust cloud of uniform
density
ρ
. A planet of mass
m
orbits around the star in a circular orbit of radius
r
within the cloud. Show that the period of the planet's orbit is given by
2
π
1
/
2
r
3
T
=
G
M
3
πρr
3
.
4
+
