I Solving for the trajectory of the center of mass

AI Thread Summary
The discussion focuses on calculating the combined translational and rotational motion of a free square with mass M in the xy plane when a linear impulse J is applied above its center of mass (CM). It is acknowledged that the impulse creates both translational motion and an angular impulse, resulting in clockwise rotation due to the torque generated. The linear motion can be determined using momentum conservation principles, while the rotational motion requires consideration of the system's overall angular momentum relative to the CM. The challenge lies in accurately computing the rotational dynamics alongside the translational effects. Understanding these interactions is crucial for developing an effective physics engine in the game.
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I'm working on the physics engine component of a game engine I'm building, and I need some guidance with this particular situation.

Consider a square with mass M that is free to translate in the xy plane and free to rotate about any axis perpendicular to the page (Fig. 1)

If a linear impulse J is applied at a point above the center of mass (CM) as shown below, I know there must be some angular impulse (momentary torque) generated since there is a component of J that is perpendicular to the displacement vector from CM. I imagine this angular impulse will tend to rotate the square clockwise.

However, I can also imagine that the CM will also undergo translation since the square is not constrained. How would I go about computing the overall rotational + translational motion of this system?
20220628_081100.jpg
 
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The linear motion is the easiest since overall momentum conservation gives
$$
\frac{d\vec p}{dt} = m\dot{\vec v}_{com} = \vec F_{tot}
$$
The rotational part can be slightly trickier due to the overall acceleration of the system. You should be able to do it by overall angular momentum, preferably relative to the CoM in the comoving frame.
 
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