A rigid spacecraft is modeled by the solid cylinder Bwhich 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?

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Accepted Answer

For the cylindrical shell A, we have
[latex]\begin{matrix} r_{B}=0.5 m & l_{B}=1.0 m & m_{B}=300 kg\end{matrix}[/latex]

The principal moments of inertia about the center of mass are found in Figure 10.9(b),
[latex]I_{B_{x}} =\frac{1}{4}m_{B}r^{2}_{B}+\frac{1}{12}m_{B}l^{2}_{B}= 43.75 kg·m^{2}[/latex]
[latex]I_{B_{y}} = I_{B_{xx}} = 43.75 kg·m^{2}[/latex]
[latex]I_{B_{z}} =\frac{1}{2}m_{B}r^{2}_{B}=37.5 kg·m^{2}[/latex]
The properties of the transverse rod are
[latex]\begin{matrix} l_{R}=1.0m&m_{R}=30 Kg\end{matrix} [/latex]
Figure 10.9(a), with r = 0, yields the moments of inertia,
[latex]I_{R_{y}} = 0[/latex]
[latex]I_{R_{z}} = I_{R_{x}} =\frac{1}{12}m_{A}r^{2}_{A} = 10.0 kg·m^{2}[/latex]
The moments of inertia of the assembly is the sum of the moments of inertia of the cylinder and the rod,
[latex]I_{x} = I_{B_{x}} + I_{R_{x}} = 53.75 kg·m^{2}[/latex]
[latex]I_{y} = I_{B_{y}} + I_{R_{y}} = 43.75 kg·m^{2}[/latex]
[latex]I_{z} = I_{B_{z}} + I_{R_{z}} = 47.50 kg·m^{2}[/latex]
Since [latex] I_{z}[/latex] 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.

Question 10.5: A rigid spacecraft is modeled by the solid cylinder Bwhich h...

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