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Principal Axes and Moments in Rotational Symmetry
Chapter 10, Problem 10.31(choose chapter or problem)
Consider an arbitrary rigid body with an axis of rotational symmetry, which we'll call \(\hat{\mathbf{z}}\).
(a) Prove that the axis of symmetry is a principal axis.
(b) Prove that any two directions \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\) perpendicular to \(\hat{\mathbf{z}}\) and each other are also principal axes.
(c) Prove that the principal moments corresponding to these two axes are equal: \(\lambda_{1}=\lambda_{2}\).
Questions & Answers
QUESTION:
Consider an arbitrary rigid body with an axis of rotational symmetry, which we'll call \(\hat{\mathbf{z}}\).
(a) Prove that the axis of symmetry is a principal axis.
(b) Prove that any two directions \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\) perpendicular to \(\hat{\mathbf{z}}\) and each other are also principal axes.
(c) Prove that the principal moments corresponding to these two axes are equal: \(\lambda_{1}=\lambda_{2}\).
ANSWER:Step 1 of 3
(a) If the 
Therefore, If \(\omega=(0,0, \omega)\), then \(\mathbf{L}=\left(I_{x z} \omega, I_{y z} \omega, I_{z z} \omega\right)=\left(0,0, I_{z z} \omega\right)\) and \(\hat{\mathbf{Z}}\) is a principal axis.
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Principal Axes and Moments in Rotational Symmetry
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Unravel the mysteries of rotational symmetry and principal axes in this video. We prove the relationships between the axis of symmetry and principal axes, showcasing the equal principal moments. Join us for an exploration of the fundamental concepts of rigid body dynamics.