Question 10.5: A rigid spacecraft is modeled by the solid cylinder B which ...
A rigid spacecraft is modeled by the solid cylinder B which has a mass of 300 kg and the slender rod R which passes through the cylinder and has a mass of 30 kg. Which of the principal axes x, y, z can be an axis about which stable torque-free rotation can occur?
For the cylindrical shell A, we have
r_{B}=0.5 m \quad l_{B}=1.0 m \quad m_{B}=300 kgThe principal moments of inertia about the center of mass are found in Figure 9.9(b),
I_{B x}=\frac{1}{4} m_{B} r_{B}^{2}+\frac{1}{12} m_{B} l_{B}^{2}=43.75 kg \cdot m ^{2}
I_{B y}=I_{B x x}=43.75 kg \cdot m ^{2}
I_{B z}=\frac{1}{2} m_{B} r_{B}^{2}=37.5 kg \cdot m ^{2}
The properties of the transverse rod are
l_{R}=1.0 m \quad m_{R}=30 kgFigure 9.9(a), with r = 0, yields the moments of inertia,
I_{R y}=0
I_{R z}=I_{R x}=\frac{1}{12} m_{A} r_{A}^{2}=10.0 kg \cdot m ^{2}
The moments of inertia of the assembly is the sum of the moments of inertia of the cylinder and the rod,
I_{x}=I_{B x}+I_{R x}=53.75 kg \cdot m ^{2}
I_{y}=I_{B y}+I_{R y}=43.75 kg \cdot m ^{2}
I_{z}=I_{B z}+I_{R z}=47.50 kg \cdot m ^{2}
Since I_{z} is the intermediate mass moment of inertia, rotation about the z axis is unstable. With energy dissipation, rotation is stable in the long term only about the major axis, which in this case is the x axis.
