Can Newton's second law for rotation be derived from Newton's second law for translation? [duplicate]

Question

I am referring to the Newton's second law for rotation:

$\vec{\tau}=I \vec{\alpha}$

Which obviously is the analogue for Newton's second law for translation

$\vec{F}=M \vec{a}$

I wonder whether the rotation law can be derived from transnational one? If yes, how can we do it?

Answer 1

Mathematically yes (as shown below), but it is always good to understand Force equations as conservation of linear momentum or angular momentum.

Taking cross-product on either side of linear force balance with $\vec{r}$, we get \begin{equation} \vec{F}\times \vec{r} = [M\vec{a}]\times \vec{r} \end{equation}

we know that, \begin{equation} \vec{F}\times \vec{r} = \vec{\tau}; \ \vec{a}= r\vec{\alpha}; \ Mr^2 = I; \end{equation}

Therefore, \begin{equation} \vec{F}\times \vec{r} = \vec{\tau} = [Mr^2\vec{\alpha}] = I\vec{\alpha} \end{equation}