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I understand when a force is applied to a body it causes moments on all points not lying on the force's line of action. So which of these points acts as the pivot point around which the body actually rotates with a specific moment as a result of that applied force?

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The motion caused by an applied force $F$ can be resolved into 2 components :

  1. A linear acceleration $a=F/m$ of the centre of mass (CM). If the force is an impulse the effect is to change the linear momentum of the CM.

  2. A rotational acceleration about the CM. The force produces a torque $\tau=r\times F$ about the CM, where $r$ is the position vector of the point of application of the force, relative to the CM. This torque causes rotational acceleration $\alpha=\tau/J$ where $J$ is the moment of inertia.

The overall motion is a combination of these 2 motions. The result usually means that there is one point in the body for which the instantaneous velocity is zero at some instant in time. This is called the instantaneous centre of rotation.

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