centre of mass

centre of mass

Let

ν

be a Borel

signed measure

for some set

E

⊆

ℝ

n

.
The

centre of mass

of

E

(with respect to

ν

) is a vector in

ℝ

n

defined by the following vector-valued

integral

:

𝝁

⁢

(

E

;

ν

)

=

1

ν

⁢

(

E

)

⁢

∫

𝐫

∈

E

𝐫

⁢

𝑑

ν

,

ν

⁢

(

E

)

≠

0

,

±

∞

,

provided the integral exists. Intuitively, the integral is a

weighted average

of all the points of

E

.

Our abstract definition encompasses many situations.
If

E

is a

k

-dimensional manifold, then we may take

ν

to be its
absolute

k

-dimensional volume element (

d

⁢

ν

=

d

⁢

V

).
This would give the set

E

a unit mass

density

,
and in this case the centre of mass of

E

is also called the

centroid

of

E

.

More generally, if we are given a

measurable

mass density

ρ

:

E

→

ℝ

,
then taking

d

⁢

ν

=

ρ

⁢

d

⁢

V

, the definition defines the centre of mass for

E

with a mass density

ρ

.
We do not restrict the density

ρ

(or the

measure

ν

) to be non-negative;
for this allows our definition to apply even to, for example, electrical charge densities.

If

E

is not a differentiable manifold, but is a rectifiable set, we can replace

d

⁢

V

by the appropriate

Hausdorff measure

.

The measure

ν

could also be discrete. For example,

E

could be a

finite set

of points

{

x

i

}

⊂

ℝ

n

with masses

m

i

=

ν

⁢

(

x

i

)

. In this case, the integral

𝝁

⁢

(

E

;

ν

)

reduces to a finite summation.

It is possible to economize the definition so that the set

E

does not have to be mentioned,
although the result is somewhat unintuitive:
the

centre of mass of a finite signed

Borel measure

ν

on

ℝ

n

is defined by

𝝁

⁢

(

ν

)

=

𝝁

⁢

(

ℝ

n

;

ν

)

=

1

ν

⁢

(

ℝ

n

)

⁢

∫

𝐫

∈

ℝ

n

𝐫

⁢

𝑑

ν

(provided this exists).
If

ν

E

is defined by

ν

E

⁢

(

S

)

=

ν

⁢

(

S

∩

E

)

for all measurable

S

and some fixed measurable

E

, then

𝝁

⁢

(

ν

E

)

=

𝝁

⁢

(

E

;

ν

)

.

The term

centre of gravity

is sometimes
used loosely as a synonym for the centre of mass, but these are distinct

concepts

;
the centre of gravity is supposed to be a single point on which the force of gravity can
be considered to act upon. If

E

is three-dimensional and the force of gravity is uniform throughout

E

, then the centre of mass and the centre of gravity coincide.

As shown by the above examples, the centre of mass and its related concepts
have applications in mechanics and other areas of physics.

Symmetry principle

If

T

:

ℝ

n

→

ℝ

n

is an

invertible linear map

, then by a straightforward change of variable
in the integrand,

𝝁

⁢

(

T

⁢

E

;

ν

∘

T

-

1

)

=

T

⁢

(

𝝁

⁢

(

E

;

ν

)

)

.

In particular, take

T

to be an isometry, and

ν

to be a

k

-dimensional volume measure for

E

.
The corresponding

k

-dimensional volume measure for

T

⁢

E

must be the same as

ν

∘

T

-

1

, because

T

is an isometry.
Then the above equation says that
the centroid transforms as expected under isometries.

For example, it is intuitively obvious that the centre of mass of a disk

D

⊂

ℝ

2

should be the centre of

D

. (Without loss of generality, assume that

D

is centered at the origin.)
Using the property just mentioned, this is easy to prove rigorously too:
if

T

is the isometry that

reflects

across the y-axis, and

λ

is the

Lebesgue measure

on

ℝ

2

,
then

T

⁢

(

𝝁

⁢

(

D

;

λ

)

)

=

𝝁

⁢

(

T

⁢

D

;

λ

∘

T

-

1

)

=

𝝁

⁢

(

T

⁢

D

;

λ

)

(

λ

is

invariant

under isometries)

=

𝝁

⁢

(

D

;

λ

)

,

(

T

⁢

D

=

D

by

symmetry

of the disk)

which means the x

component

of

𝝁

⁢

(

D

;

λ

)

must be zero. Similarly, by considering

reflections

across the x-axis, we conclude
that the y component of

𝝁

⁢

(

D

;

λ

)

must be zero.

Title

centre of mass

Canonical name

CentreOfMass

Date of creation

2013-03-22 15:28:09

Last modified on

2013-03-22 15:28:09

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

12

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 28A75

Classification

msc 26B15

Synonym

center of mass

Related topic

PappussTheoremForSurfacesOfRevolution

Defines

centre of gravity

Defines

centroid

Defines

center of gravity

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