Second Law for Rotational Motion Moment of inertia is analogous to mass, and angular acceleration is analogous to linear acceleration. What is analogous quantity to net force? In other words, what is moment of inertia*angular acceleration?
 A: John Rennie's answer is correct in the special case that angular momentum and angular velocity are parallel to one another. This is not always the case as moment of inertia is a second order tensor.
Angular momentum is given by $\boldsymbol L = \boldsymbol{\mathsf{I}} \boldsymbol \omega$. Differentiating this gives the rotational analog of Newton's second law: $\boldsymbol \tau = \dot{\boldsymbol L} = \dot{\boldsymbol{\mathsf{I}}} \boldsymbol \omega + \boldsymbol{\mathsf{I}} \dot{\boldsymbol \omega}$. The first term on the right hand side, $\dot{\boldsymbol{\mathsf{I}}} \boldsymbol \omega$, is a challenge.  The moment of inertia tensor of a rotating rigid body is non-constant from the perspective of an inertial frame.
A way around this is to look at things from the perspective of a frame fixed with respect to the rigid body. The time derivatives of a vector quantity $\boldsymbol q$ from the perspective of instantaneously co-moving inertial and rotating frames are related via
$$\left(\frac{d \boldsymbol q}{dt}\right)_{\text{inertial}} = \left(\frac{d \boldsymbol q}{dt}\right)_{\text{rotating}} + \boldsymbol \omega \times \boldsymbol q$$
Applying the above to $\boldsymbol L = \boldsymbol{\mathsf{I}} \boldsymbol \omega$ yields
$$\begin{aligned}
\boldsymbol \tau_{\text{ext}}
   = \left(\frac{d \boldsymbol L} {dt}\right)_{\text{inertial}}
  &= \left(\frac{d \boldsymbol L}{dt}\right)_{\text{rotating}}
    + \boldsymbol \omega \times \boldsymbol L \\
  &= \left(
      \frac{d\, \boldsymbol {\mathsf{I}} \boldsymbol \omega}{dt}
     \right)_{\text{rotating}}
    + \boldsymbol \omega \times (\boldsymbol{\mathsf{I}} \boldsymbol \omega) \\
  &= \boldsymbol {\mathsf{I}} \dot {\boldsymbol \omega}
    + \boldsymbol \omega \times (\boldsymbol{\mathsf{I}} \boldsymbol \omega)
\end{aligned}$$
This is Euler's equation for rigid body dynamics. The first term on the right hand side is John Rennie's answer. The second term is the rotational analog of a fictitious force.
A: The equivalent to force is torque, $\tau$. So the second law would be:
$$ \tau = I\dot{\omega} = I\ddot{\theta} $$
