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When writing the equations for center of mass we assume as it to be the point where all the forces act and write the corresponding kinematic equations, and in doing so we dont assume any distribution of forces (like constant or linear wrt length or any dimension of the body)

But when we consider the case of a rod lying on a surface with a force acting as F(x) = Kx where x is the distance from one end of the rod, we see that the force is more concentrated above the center of mass and the net force doesnt necessarily pass through the com. I am unable to wrap my head around this difference in result. Need some help understanding where im going wrong in my thinking

Qmechanic
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abhiyeet
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  • The offending statement that talks about "where all the forces act", it really is "appears to act" and referring to centre of gravity, not mass, and only one type of force at a time. We define things this way because it is clearly a useful mathematical abstraction of great practical use. When you have your weird distribution of forces, that is not a typical case, and the concept of centre of mass would not be useful in your problem. – naturallyInconsistent May 12 '23 at 07:08
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    If all the forces really always acted at the center of mass, they could never cause rotation around the center of mass. – Ghoster May 12 '23 at 07:23
  • That all forces are acting only on the center of mass is the so called "particle approximation". It works well in some problems, like the description of the orbit of a planet around the sun and it is completely useless in others, e.g. if we want to describe the motion of a plane or the deformation of a loaded beam. If you want to have fun in physics, then you have to learn to live with many such approximations because that is what physics is: the art of approximation. – FlatterMann May 12 '23 at 07:51

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Any force that acts along a line that does not pass through the centre of mass can be shown to be equivalent to a force of equal magnitude acting through the centre of mass plus a rotational couple about the centre of mass. The couple does not affect the motion of the centre of mass itself, so we can ignore it when analysing how the centre of mass moves. This why when deriving equations of motion for the COM, we can treat all forces as if they acted through the COM.

In the case of the rod, however, we are interested not just in the motion of its COM, but also in its orientation, which is affected by the couple. So in this case we cannot just replace forces by equal forces acting through the COM.

gandalf61
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