Solutions Assignment 9
Verify the relations in (8.52) The semimajor axis is found from
ro
2a = rmax + rmin =
a =
+
1
ro
ro
1+
=
2ro
2
1
2
1
The semiminor axis is found from
b = ymax :
Since y = r sin
We …nd
dy
d
dy
d
the maximum value for y is determined by setting dy=d = 0:
sin
d
= ro
d 1 + cos
= ro
cos
1 + cos
= ro
cos (1 + cos ) +
1
sin2
+
2
(1 + cos )
!
cos2
= ro
2
(1 + cos )
Setting this expression to zero yields cos o =
p
ro sin o
ro 1
ymax =
=
2
1 + cos o
1
!
cos +
: Hence
2
=p
ro
1
2
=b
The o¤set, d, is simply
d
d
= rmax
=
ro
1
a=
1
ro
ro
1
1
1+
2
1
=
ro
1
2
= a:
8.19 At perigee and apogee respectively,
rp = y p + R e =
`2 =GMe
`2 =GMe
; and ra = ya + Re =
:
1+
1
Solving for the eccentricity we …nd
rp
ra
=
=
=
1
! (1 + ) rp =ra = 1
1+
1 rp =ra
ra r p
ya y p
=
=
1 + rp =ra
ra + r p
ya + yp + 2Re
2700
= :1677
3300 + 2 6400
1
2
(1 + cos )
!
:
From the statement of the problem, as the satellite crosses the y axis
and
ry
y
y
= =2
= y + Re = `2 =GMe = (1 + ) (yp + Re )
= (1 + ) (yp + Re ) Re = (1 + ) yp + Re
= 1:1677 300 + :1677 6400 = 1424km
8.20 The expression for the orbit is
r( ) =
`2 =GM
:
1 + cos
At apogee
`2 =GM
! rmax (1
) = `2 =GM :
1
Therefore if we hold rmax …xed and let ` ! 0 then must approach 1. For = 1;
rmax =
rmin =
`2 =GM
1+
= `2 =2GM :
As ` ! 0; rmin ! 0 as well. With this analysis it is clear that if rmax is …xed
and ` is small then the eccentricity is close to 1. This is an orbit for which the
semimajor axis is much larger than the semiminor axis. The semimajor axis is
expressed as
2a = rmax + rmin =
For
`2 =GM
1
+
`2 =GM
1+
= rmax +
1
2
rmax = rmax
:
1+
1+
very close to 1, a ' rmax =2:
8.21 (b) Kepler’s third law states
2
=
4 2 3
a :
GM
For the case described in 8.20, a ' rmax =2. Hence in terms of rmax Kepler’s
third law becomes
2
2
=
2GM
3
rmax
:
(c) For this orbit the total time to fall from r = rmax (where its total
energy is just its potential energy) is
r Z 0
r Z 0
dr
dr
p
p
=
T =
2 rmax U (rmax ) U (r)
2 rmax
G M =rmax + G M =r
s
s
p
Z 0
Z 0
1
dr
1
rdr
p
p
T =
=
:
2GM
2GM
1=rmax + 1=r
1 r=rmax
rmax
rmax
2
The minus sign is used as the radial velocity is negative (the radius is decreasing).
De…ning
r = rmax cos2 ;
the integral becomes
s
s
Z =2
Z
1
cos (2 sin cos d )
2
3=2
3=2
rmax
=
rmax
T =
2GM
sin
GM
0
0
s
s
2
1
3=2
3=2
T =
rmax
=
rmax
:
GM
4
8GM
=2
cos2 d
This comet approaches the Sun on a nearly radial line. As it reaches the Sun it
makes a U turn and returns.
(d,e) The period for this orbit is
s
= 2T =
1
2GM
3=2
rmax
:
Squaring both sides yields
2
2
=
2GM
3
rmax
;
which is in exact agreement with part (b).
8.23 (a,b) The potential energy for a particle of mass m in the force …eld
k
+ 3
r2
r
F (r) =
is
k
+ 2:
r
2r
If the particle moves with an angular momentum L then the expression for the
conservation of energy is
U (r) =
E
=
E
=
1 2
L2
1 2
mr + U (r) +
= mr
2
2mr2
2
1 2 k L2 + m
mr
+
:
2
r
2mr2
k
L2
+ 2+
r
2r
2mr2
As usual we r = 1=u or equivalently u = 1=r: Also we note that d=dt = d=d :
First consider the radial kinetic energy term
r
r
2
=
=
dr du
=
du dt
L
m
2
1 du
=
u2 d
du
d
2
:
3
Lu2 1 du
=
m u2 d
L du
md
Substituting this result into the conservation of energy in the special case where
k = 0 we …nd
L2
2m
2
du
d
L2 + m 2
u
2m
+
= E;
2
du
d
+ 1 + m =L2 u2
1
1 + m =L2
=
2m
E;
L2
=
2m
E:
L2 + m
2
du
d
+ u2
From observation the solution to this nonlinear di¤erential equation is
u = u0 cos
; with
=
q
p
p
1 + m =L2 and u0 = 2mE= (L2 + m ) = 2mE=
With k nonzero the expression for the conservation of energy is
L2
2m
2
du
d
+
L2 + m 2
u
2m
ku = E
Going through the same procedure as that with k = 0 we …nd
1
1 + m =L2
du
d
1
2
+ u2
=
2mk
2 2u
L
=
2
du
d
2
2mk
u
+m
L2
+ u2
L2
2m
E
+m
2m
2 2 E:
L
Completing the square results in
1
2
2
du
d
m2 k 2
2mk
u
+
2 2
4 4
L
L
+ u2
1
2
du
d
2
mk
2 2
L
+ u
=
2m
m2 k 2
E
+
2 2
4 4;
L
L
=
2m
m2 k 2
2 2E + 4 4 :
L
L
2
From the result for the case with k = 0 we see that the solution is now
s
mk
2m
m2 k 2
u =
+
E
+
;
2 2
2 2
4 4 cos
L
L
L
q
mk
u =
1
+
1 + 2 2 L2 E=mk 2 cos
;
2 2
L
where again
=
Inverting this expression we …nd
p
1 + m =L2 :
2
r( ) =
q
1+ 1+2
4
L2 =mk
2
L2 E=mk 2 cos
2
L2 :
This is of the form
c
1 + cos
r( ) =
where
2
c=
as
L2 =mk and
=
;
q
1+2
2
L2 E=mk 2 :
(c) This orbit is closed whenever = n=m; a rational number. Note that
! 0 the parameter ! 1 and the solution is that for a Kepler orbit.
8.29 The kinetic energy of the Earth would remain unchanged. However
the potential energy would immediately be halved. In a circular orbit the virial
applies not just on average but for all time. Hence prior to the Sun losing its
mass
1
n
U; and E = T + U = U=2:
T = U=
2
2
Since U is negative this is the energy of a bound particle. If the Sun lost half its
mass then relative to its new potential energy T = U: Now the total energy is
E = T + U = 0:
The Earth is just barely unbound.
8.35 Assume that the initial radius for the circular orbit is R1 . After a
backward thrust given by = v2 =v1 < 1; the orbit will become an ellipse with
the rocket located at the apogee. Hence
R1 = `21 =GM =
2 2
`1 =GM
`22 =GM
=
1
2
1
2
=
2
1
R1 :
2
This implies that
2
=1
2
!
2
2
=1
:
At the perigee the distance from the Sun is R3 given by
R3 =
Solving for
2
2 2
`1 =GM
2
`22 =GM
=
1+ 2
2
2
=
2
2 R1 :
we …nd
2
2
R3 =
2
R1 !
2
=
2R3
:
R1 + R 3
p
Since R3 = R1 =4; = 2=5 = 0:6325:
To obtain a circular orbit at this radius an additional backward thrust is
required at the perigee. Since R3 is held …xed we …nd
R3
=
02
=
`22 =GM
= `23 =GM = 02 `22 =GM
1+ 2
1
1
R1 + R 3
=
:
2 =
1+ 2
2R1
2
5
Again R3 = R1 =4 so that
The …nal velocity is
v3
v3
11.6
0
=
p
5=8 = 0:7906:
(per)
`2 =R3
v1 = 0
v1 =
v2 (apo)
`2 =R1
p
=
R1 =R3 v1 = 2v1 :
=
0 v2
r
R 1 + R3 R1
2R1 R3
r
2R3
v1
R 1 + R3
(a) The Lagrangian for this system (m1 = m2 = m; k1 = 3k; and k2 = 2k)
is
L=
2
2
1
m x1 + x2
2
1
2k (x2
2
1
3kx21
2
2
x1 ) :
Hence the equations of motion are
mx1
=
3kx1
2k (x1
mx2
=
2k (x2
x1 )
x2 ) =
5kx1 + 2kx2
Assuming a solution of the form
z=
a1 i!t
e
a2
where x = Re z; we …nd
5k
m! 2
2k
2k
2k
m! 2
a1
0
=
:
a2
0
A nontrivial solution requires the secular equation,
det
5k
m! 2
2k
2k
2k
= m2 ! 4
m! 2
7km! 2 + 6k 2 = 0;
to be satis…ed. The normal mode frequencies are
! 21 = k=m; and ! 22 = 6k=m:
(b) To …nd the ratios of a1 and a2 for ! 1 we …nd
5
1
2
2
1
a1
=
a2
4
2
2
1
a1
= 0:
a2
Hence
2a1 = a2 :
In this mode the oscillations are in phase with the amplitude of x2 being twice
that of x1 :
6
To …nd the ratios of a1 and a2 for ! 1 we …nd
5
6
2
2
4
a1
=
a2
1
2
a1 =
2a2 :
2
4
a1
= 0:
a2
Hence
In this mode the oscillations are exactly out of phase with the amplitude of x1
being twice that of x2 :
The equations of motion when m1 = m2 = m and k1 = k2 =
11.9 (a)
k3 = k are
mx1
=
2kx1 + kx2
mx2
=
2kx2 + kx1 :
The normal coordinates are
1
= (x1 + x2 ) =2 and
2
= (x1
x2 ) =2:
Hence
x1 =
1
+
2
and x2 =
2:
1
Substituting this result into the equations of motion leads to
m
1
m
1
+
2
=
2k (
1
2
=
2k (
1
+
2)
+k(
1
2)
+k(
1
+
2)
=
2)
=:
k
3k
1
k
1
2
+ 3k
2
Adding and subtracting these two expressions results in
m
1
=
k 1;
m
2
=
3k 2 :
(b) The solutions are
1
= A1 cos (! 1 t
1)
and
2
= A2 cos (! 2 t
2)
where ! 21 = k=m and ! 22 = 3k=m: The general solution for the displacements is
x1 (t)
=
x2 (t)
x (t)
=
x (t)
= A1
(t) +
1 (t)
1
1
cos (! 1 t
1
1)
7
(t)
2 (t)
2
+ A2
1
cos (! 2 t
1
2) :