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When any body like a pen is given a gentle hit why does it rotate about its center of mass?

I gave my pen a hit from left end and executed circular motion about its center of mass? Why is it so?

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It doesn't have to rotate about its COM during the push. But once you let go there is no net force acting on the body. Therefore, the COM cannot be accelerating. The solution to $\ddot{\mathbf x}=\mathbf a=0$ is $\mathbf x(t)=\mathbf v_0t+\mathbf x_0$, where $\mathbf x_0$ and $\mathbf v_0$ are the position and velocity respectively of the center of mass when the push stops. So the COM will just move in a straight line while the other points of the body rotate about the COM.

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I apparently looks so to you because you are trying it on a rough surface where friction is giving a force opposite to your force, equal in magnitude and as a result there is no net linear acceleration.

It does not rotate about center of mass always. There are two things going on. First, find linear acceleration of center of mass by simple Newton's second law F=ma. Second, find the angular acceleration of rod (or pen) by the torque equation tau = I*alpha about the center of mass. Although, center of mass is an non-inertial frame to calculate alpha, still we use it; to compensate the effect, we will subtract or add linear acceleration F/m as per required to nullify the effect. Now, it is very easy to calculate acceleration of left and right extreme ends and accordingly find instantaneous axis of rotation.

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