Discussion
Euler's equation of motion for a rigid body about a fixed point in the absence of any net torque and \(I_{xx} = I_{yy}\) imply that the \(z\)-component of the angular velocity :
Answer: B
The \(z\)-component of the angular velocity is constant.
Euler's torque-free equation for the third principal axis is
\[ \begin{aligned} I_{zz}\dot{\omega}_{z} &= (I_{xx} - I_{yy}) \\ &\quad \times \omega_{x}\omega_{y} \end{aligned} \]
With \(I_{xx} = I_{yy}\) the right side vanishes, so \(\dot{\omega}_{z} = 0\) and \(\omega_{z}\) is constant.
It is the other two components that perform simple harmonic motion (they precess about the symmetry axis), so option A describes \(\omega_{x}\) and \(\omega_{y}\), not \(\omega_{z}\). \(\omega_{z}\) need not be zero, and it does not vary with time.
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