How to resolve impulse on a free-floating body into translation and rotation

The general motion of a planar rigid body is a rotation about a point. If you consider an impulse $hat{j}$ that passes on a line a distance $a$ from the center of mass, then the center of rotation is going to be a distance $b$ on the other side of the center of mass as seen below

The relationship between the two distances is

$$ b = frac{I}{m a} tag{1}$$

or if you use the radius of gyration $r$, and use $I = m r^2$ then

$$ b = frac{r^2}{a} tag{2}$$

For example for a rectangle of length $ell$ and height $h$ the MMOI is $ I = tfrac{m}{12} ( ell^2 + h^2 )$, or $r = tfrac{sqrt{3}}{6} sqrt{ ell^2+h^2}$

The great thing about the above relationship is that it is purely geometrical. It also has the property that the less $a$ is the larger $b$ is, and vice versa.

There are three special cases to (2)

• $a=0$, the impulse goes through the center of mass, and the body translates since $b=infty$.
• $b=0$, the body is rotating about the center of mass when $hat{j}=0$ and $a=infty$. This situation corresponds to a pure impulsive torque on the body.
• $a=r$ the impulse is tangential to the circle of gyration, then $b=a$.