Q: Why does the axis of rotation of a spinning top tilt further away from the vertical after some time? (Neglect air friction. Assume real-life situation otherwise.)
A: Friction at the pivot point drains the top's rotational kinetic energy, causing its spin speed to drop over time. The spin angular momentum $L=Iω$ therefore decreases as the top slows down. Reduced angular momentum means less resistance to a change in the direction of the axis.
In a real situation, there are always small moments when the axis is not absolutely vertical. In such instances, gravity exerts a torque that tends to tip the top over. The top responds to this torque by precessing—the axis rotates around the vertical. As the spin speed decreases, the rate of precession increases and the motion becomes increasingly pronounced.
As the top tilts farther, the gravitational torque also increases because the perpendicular distance from the pivot to the line of action of gravity increases. With less angular momentum available to resist the resulting change in the direction of the axis, the tilt progressively increases. Friction continues to dissipate energy, and eventually the top's axis moves farther and farther from the vertical until the top falls over.