# Determining the angular and linear accelerations of a Rigid Body

1. Aug 9, 2009

### Arrowstar

Hello, all!

I am currently working with the governing equation of a rotating, translating rigid body. That is:

$$\vec{a_i} = \vec{a_c} + \vec{\alpha} \times \vec{r} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r}} \right)$$

Where $$a_i$$ is the linear acceleration of some point on the body $$i$$, $$a_c$$ is the linear acceleration of the centroid of the body, $$\alpha$$ is the angular acceleration of the body, $$\omega$$ is the angular velocity of the body, and $$r$$ is the vector from the centroid to the point $$i$$.

Given some $$a_1$$ and $$a_2$$, how can I rework a system of two of those equations such that I can back out $$\alpha$$ (related to omega by the differential) and $$a_c$$? That is:
$$\vec{a_1} = \vec{a_c} + \vec{\alpha} \times \vec{r_1} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_1}} \right)$$
$$\vec{a_2} = \vec{a_c} + \vec{\alpha} \times \vec{r_2} + \vec{\omega} \times \left( {\vec{\omega} \times \vec{r_2}} \right)$$

To give another way of viewing the problem: If I mount two accelerometers on a body and then rotate and translate the body in space, how can I determine what those body rotations and translations are based off the readings of my two accelerometers? I should only need two accelerometers (read: two equations) because there are only two unknowns, $$a_c$$ and $$\alpha$$.

Thank you for the help!