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Physical Motivation for Center of Mass Formulas using Torque Equations, Study Guides, Projects, Research of Calculus

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.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

mayer
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Centroids and moments
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
bounded planar object whose center of mass (or centroid) has coordinates
(x*, y*),
and choose positive numbers
a
and
b
such that all points on A belong to the solid
rectangular region B defined by the following inequalities:
x* – a
x
x* + a
y* – b
y
y* + b
Think of B as a flat, firm rectangular sheet with uniform density (made of glass, metal,
wood, plastic,
etc
.) such that A rests on top of B.
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
the edge E of C along the vertical line defined by the equation
x = x*
.
pf3

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Centroids and moments

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

bounded planar object whose center of mass (or centroid) has coordinates ( x *, y *),

and choose positive numbers a and b such that all points on A belong to the solid

rectangular region B defined by the following inequalities :

x * – a ≤ x ≤ x * + a

y * – b ≤ y ≤ y * + b

Think of B as a flat , firm rectangular sheet with uniform density (made of glass , metal ,

wood , plastic , etc. ) such that A rests on top of B.

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

the edge E of C along the vertical line defined by the equation x = x *.

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

line defined by the equation x = x * is equal to the total torque or moment of A to the

left of the vertical line defined by the equation x = x *.

If the mass distribution on A is given by the (continuous) function ρ

ρ ( x , y ), then the

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 :