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Consider a body which is static with respect to an inertial reference frame. If I apply an impulse $\vec{F} dt$ with known magnitude and direction at a point $P$ which is different than the center of mass of the body, how could I know the body's future positions, velocity and angular velocity? Does will the body rotate around of its center of mass?

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If an external impulse is applied to an object (or a system), there will be a corresponding change in the linear momentum (which can be expressed in terms of the motion of the center of mass). The impulse multiplied by the distance between the center of mass and the line of action of the impulse, will also produce an angular impulse. This will produce a corresponding change in the angular momentum about the center of mass. An object rotates about the center of mass because any other motion would require an externally applied centripetal force.

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