Solution
= Solution
For $u=0$, the equation becomes
$$
\frac{\dot\omega}{\omega}=\frac{\dot m}{m},
$$
so $\omega/m$ is constant. With $m(t)=m_0-\mu t$,
$$
\omega(t)=\omega_0\frac{m_0-\mu t}{m_0}.
$$
It reaches zero when the water is exhausted, at
$$
\boxed{t_{\rm stop}=\frac{m_0}{\mu}}.
$$