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This note provides physical motivation for the standard formulas of the center of mass of an object in the plane using torque equations. It discusses the torque equation for a continuous mass distribution and a discrete mass distribution, and how they lead to the standard formulas for the center of mass. It also shows the application to barycentric coordinates.
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The purpose of this note is to provide some physical motivation for the standard
formulas which give the center of mass of an object in the plane. Suppose A is a
rectangular region B defined by the following inequalities :
Think of B as a flat , firm rectangular sheet with uniform density (made of glass , metal ,
Next , suppose that we have a triangular rod C with equilateral ends , positioned so that
one of the lateral faces is horizontal and the opposite edge E lies above this face.
As suggested by the figure on the next page , suppose that we now we rest B and A on
Since we are resting A and B along a line containing the center of mass for this
combined physical system of objects , we expect that the combined object will
balance perfectly , not tipping either to the left or right.
Physically , this means that the total torque or moment of A to the right of the vertical
torque equation is given by the following integral formula :
( x ***** x ) ( x , y ) dxdy ( x x *) ( x , y ) dx dy
LEFT RIGHT
−−−− ρρρρ ==== −−−− ρρρρ
This equation can be rewritten in the following standard form :