A wheel spins about its symmetry axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Index notation and Cartesian tensors

Problem

A wheel spins about its symmetry axis, taken along zz, with angular velocity ω=(0,0,150) rad s−1\boldsymbol\omega = (0, 0, 150)\,\mathrm{rad\,s^{-1}}; the wheel's disc lies in the xyxy plane. A plane mirror coincides with the xzxz plane, so the reflection sends (x,y,z)→(x,−y,z)(x,y,z)\to(x,-y,z). ((a)) A quick argument says: "the mirror doesn't touch the zz axis at all, so the zz-component of ω\boldsymbol\omega is left alone by the reflection." Using the fact that ω\boldsymbol\omega is an axial vector, find the zz-component of the angular velocity of the mirror-image wheel, ωz′\omega'_z, and say whether the quick argument is right. ((b)) State in one sentence what physical feature of ω\boldsymbol\omega (as opposed to an ordinary displacement vector) is responsible for the result.

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