The earth-scanning satellite is in a circular orbit of period . The angular velocity of the satellite

The earth-scanning satellite is in a circular orbit of period τ. The angular velocity of the satellite about its y- or pitch-axis is w = 2π/τ, and the angular rates about the x- and z-axes are zero. Thus, the x-axis of the satellite always points to the center of the earth. The satellite has a reaction-wheel attitude-control system consisting of the three wheels shown, each of which may be variably torqued by its individual motor. The angular rate Ω_z of the z-wheel relative to the satellite is Ω₀ at time t = 0, and the x- and y-wheels are at rest relative to the satellite at t = 0. Determine the axial torques M_x, M_y, and M_z which must be exerted by the motors on the shafts of their respective wheels in order that the angular velocity w of the satellite will remain constant. The moment of inertia of each reaction wheel about its axis is I. The x and z reaction-wheel speeds are harmonic functions of the time with a period equal to that of the orbit. Plot the variations of the torques and the relative wheel speeds Ω_x, Ω_y, and Ω_z as functions of the time during one orbit period.

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