(Solved): A child buys a top composed of a thin uniform cylinder and a broad uniform disk. The \( 14.2 \math ...

A child buys a top composed of a thin uniform cylinder and a broad uniform disk. The \( 14.2 \mathrm{~g} \) cylinder has a total length of \( 6.40 \mathrm{~cm} \). The \( 43.5 \mathrm{~g} \) disk has a radius of \( 2.35 \mathrm{~cm} \), and its center of mass is located a distance of \( 1.78 \mathrm{~cm} \) from the tip at the bottom of the thin cylinder. The child places the top on its pedestal and gives it an angular speed of \( 100 \mathrm{rad} / \mathrm{s} \). When she turns to show her parents, she accidentally bumps the top, which begins to precess about a vertical axis through the point where the top contacts the pedestal. As the top precesses, it makes an angle \( \phi=12.4^{\circ} \) with the vertical axis. Determine the precessional angular speed \( \omega_{\mathrm{p}} \) of the top. Use \( g=9.81 \mathrm{~m} / \mathrm{s}^{2} \) for the acceleration due to gravity, and assume that the moment of inertia of the thin cylinder is negligible.