lecture 19 centre of mass

7/28/2019 Lecture 19 Centre of Mass

1/4

LECTURE 19

DENSITY, MASS AND CENTRE OF MASS

Consider a lamina of varying composition (for example a thin sheet of metal) inthe x y plane with density (x, y) at the point (x, y). Then

Mass() = M =

(x, y)dA.

If the centre of mass of is (x, y) then

x =

x (x, y)dA

M=

My

M

and

y =

y (x, y)dA

M=

Mx

M

My =

x (x, y)dA is called the first moment of the lamina about the y axis.

Mx =

y (x, y)dA is called the first moment of the lamina about the x axis.

If the density is uniform with (x, y) = 1 then the centre of mass is referred to as acentroid. Note that the mass is then

Mass() = M =

1dA = Area().

Generally density =mass

volume, however since the lamina is assumed to have negligible

thickness we can say that density =mass

area. Thus (x, y) =

dm

dAimplying that

dm = (x, y)dA. Integrating to obtain the total mass we have

Mass() = M = (x, y)dA. That is, mass is the integral of density.

When calculating x =

x (x, y)dA

Mwe are simply taking a weighted average of

all the xs in with the values of x of higher density making a greater contribution tothe x coordinate of the center of mass. Similarly for y.

1


2/4

Other formulae of interest are:

Moments of Inertia

The moments of inertia of the above lamina about the x and y axes are

Ix =

y2(x, y)dA

Iy =

x2

(x, y)dA.

The polar moment of inertia about the origin is defined by

I0 = Ix + Iy =

(x2 + y2)(x, y)dA.

Moments of inertia measure an objects resistance to spinning about a particular axis.A spinning ice skater with her arms out has a larger moment of inertia about the verticalaxis than one with her arms tucked into her body.

Example 1 A thin plate is the region in the first quadrant bounded by the coordinateaxes and x + y = 1. Find the mass M and centre of mass (x, y) of the plate assuminga) uniform density (x, y) = 1.b) non uniform density given by (x, y) = xy.

a)12 , (13 , 13) b) 124 , (25 , 25)

2


3/4

Example 2 Find the mass, centre of mass and the moment of inertia about the y-axisof a thin plate bounded by the curves y = x2 and y = 2x x2 with variable densitygiven by (x, y) = x + 1.

M = 12 , Mx = 415 , My = 415 , (x, y) = ( 815 815), Iy = 16

3


4/4

Example 3 Find the mass and the centre of mass of a lamina bounded by y =

1 x2and the x axis assuming:

a) uniform density (x, y) = 1.

b) variable density given by (x, y) = y.

c) variable density given by (x, y) = r.

a)2

, (0, 43

) b)23

, (0, 316

) c)3

, (0, 32

)

19You can now do Q 72 to 76

4

lecture 19 centre of mass

Documents