# Why does angular momentum change only its direction and not its value (module) in the case of a spinning top?

I have a doubt, I hope you can help me. In the case of a spinning top precessing around the $y$-axis, there's a torque $\vec \tau$ which comes from the weight of the toy. This torque is perpendicular to the angular momentum $\vec L$. We have the relation: $$\frac{d\vec L}{dt} = \vec \tau$$ I understand angular momentum changes its direction, but why not the module?

This is formally equivalent to the fact, that a magnetic field does not change the magnitude of a velocity. The answer is, because the torque is perpendicular to the angular momentum (as you point out, cast in math $\vec L \cdot \vec \tau = 0$). Then it is a one liner: $$\partial_t \left|\vec L\right| = \frac{2 \vec L \cdot \dot{\vec L}}{2 \sqrt{\vec L \cdot \vec L}} = \frac 1 {\left|\vec L\right|}\vec L \cdot \vec \tau = 0.$$