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We know that if no external force is acting on a body and if a body has started moving from rest due to internal forces then the displacement of centre of mass will be zero but if we consider a case in which a body is under free fall then its centre of mass will have some displacement, what does this imply as it is still under an internal force i.e gravitational force.

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Your confusion seems to stem from considering gravitation to be an internal force. If you were to jump out of a plane, you would start falling, due (classically) to the external force of gravity acting on you, if you want to think about the system of you alone. If you want to think about the system of the Earth and you, then while you are displaced toward the Earth, the Earth is displaced toward you in exactly the amount needed to keep the center of mass stationary. Given the significant mass difference between you and the Earth though, you will be displaced much more than the Earth.

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I'm trying to imagine the setup. If an object is moving due to gravity (Freefall), then, the center of mass will move as well. However, a center of mass moving at constant speed, where the sum of all forces is zero, will still have a displacement. So the claim is that, when no external forces are applied, the center of mass will preserve its speed.

Let's say that you have a system of two objects, where one is the Earth and the other is a falling object. In this system, there are two forces, one applied to the falling object and the other to the Earth, both with the same magnitude and opposite direction. The center of mass will be then somewhere between the center of mass of both objects. Since the Force is proportional to the mass, the displacement of the center of mass of each one will be inversely proportional to the mass. In other words, the displacement of the center of mass of the Earth is so small, that nobody will notice. However, it is exactly the displacement needed to preserve the center of mass of the binary system.

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