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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...
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