## Transcribed Text

3. Satellites spin so as to keep a fixed axis; this is especially important
in maintaining communication with ground stations as antennas and
dishes have to point in a single consistent direction. A question that
needs to be asked is: is it better to have short, "fat" satellites or tall,
"slender" satellites? This is a question of stability. To understand
this, we start with the Euler equations for the rotation of a rigid body:
= 15'(12-13) 62233
dw
dt
dw2 = -
dt
dw3
df
= 15°(11-12)a2,W2- -
(a) For the two cases J1 J2 = J3 and /1 > J2 = J3 determine the
stability of w1 = W10,W2 = W3 = 0 via the linearization of the
Euler equations.
(b) On the other hand, since the stability found is only marginal sta-
bility, and satellites are not rigid bodies (but have flexible com-
ponents), another way of considering the issue is to minimize
the kinetic energy subject to a fixed angular momentum. That
is, if Jo is the moment of inertia matrix in the original orienta-
tion and w is the angular velocity in fixed co-ordinates, then
we want to find the orientation Q (a 3 X 3 orthogonal matrix)
2
that minimizes wt OT JoQw subject to Q7 JoQw = Lo where
Lo is the original (and final) angular momentum. [Hint: Put
we = Qw and retwrite the problem in terms of body-relative
angular velocity wo. Also put Jo = diag(J1 J2, J3).]
(c) How can you decide which of (a) or (b) to use? Under what
circumstances is (a) or (b) more suitable?
4. We live on a moving Earth, and the strongest effect of the Earth's mo-
tion on dynamics on the Earth's surface is due to its rotation. Sup-
pose that the angular velocity of the Earth is fixed o (which points in
the direction of the axis of rotation). Note that if xo(t) is the position
of a point in Earth-fixed co-ordinates while x(t) is the corresponding
position in space-fixed co-ordinates (philosophically: in co-ordinates
fixed relative to the distant stars). Then x(t) = Q(1)xo(t) where Q(t)
is the orientation of the Earth. Show that
dx dt = Q(t) dxo att o X
Set up the Lagrangian for a particle of mass m at position x(t)
=
Q(t)xo(t) in terms of XO and its derivatives, and find the Euler-Lagrange
equations in terms of Xo. [Notes: For a rotation matrix Q, Q(a X b) =
Qa X Qb. Also, assume that Q(1)n = o for all t as o points in the
direction of the axis of rotation.]

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Question 4

r_0 is vector and r_0=|r_0 |

We have the transformation

r(t)=Q*r_0 (t)

where r is position vector in fixed system and r_0 is position vector in...