TITLE: Balancing stick on hand - inverted pendulum motion
QUESTION [0 upvotes]: How do I calculate the velocity and acceleration of a stick that is vertically on my hand? How fast and far do I need to move my hand from side to side to keep balancing the stick? Assuming that we neglect friction force.

REPLY [1 votes]: Classically and idealised, the acceleration is of course zero. Only, there will always be the slightest deviation from the unstable equilibrium, resulting in an exponential divergence while the tilt angle is small:
$$\begin{aligned}
  \ddot{\vartheta}(t) \propto \tau \propto \sin(\vartheta) \approx& \vartheta
  & \Rightarrow \vartheta(t) \propto e^{\mu t}
\end{aligned}$$
Where $\mu$ is a constant that can be calculated from the stick's moment of inertia $I$, its center of mass' height $h$, and the gravitational acceleration $g$:
$$\begin{aligned}
  I \ddot{\vartheta} = \tau =& h\cdot g\cdot \sin(\vartheta)
    &\Rightarrow \mu^2I =& hg & \Rightarrow \mu = \sqrt{\tfrac{hg}{I}}.
\end{aligned}$$
However, that value is in a sense irrelevant: if you wait long enough, the acceleration will get "arbitrarily" large.
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