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Calculus and AnalyticalGeometry 2

Momentsand Center

of Massngcy@ucsiun

u.my

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=

k

kk

m

xmx

massofcentressystem'=x

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If the system balances at the fulcrum, distance away from orig

moment about the fulcrum must be equal to zero.

moments about fulcrum on LHS = moments about fulcrum o

!"

#"$

x

x

x

=

++

++=

++=++

++=++

=+

=+

mass

originaboutmoment

%&

%&

%&

%&%&%&

"'(

""''((

""''(("'(

""''(("'(

"""'''(((

""''((

x

mmm

xmxmxmx

xmxmxmmmmx

xmxmxmxmxmxm

xmxmxmxmxmxm

xxgmxxgmxxgm

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)or the case of constant density, ,

Moment about the origin:

Mass:

Centre of mass:M

Mx

dxM

dxxM

o

b

a

b

a

o

=

=

=

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Example 1:

The 10m long rod thickens from left to right so that its

is non constant. Find the rods center of mass.

xx

(*(%& +=

M

Mx

dxM

dxxM

o

b

a

b

a

o

=

=

=

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Center of mass

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lets suose that the late is the region boundet!o cur"es and on the inter"al #a,b$% &o, !e !antthe center of mass of the region belo!%

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Moments of Bounded Area

( constandensity

Mass(M)

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Equations of Center of ma

Mass

Moment( in terms of xsand yscoordinates)

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Center of Mass Coordinates

)he coordinates of the center of mass, , are then,

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Example

*etermine the center of mass foregion bounded by , y + 0 ointer"al % Gi"en that the density dconstant%

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*etermine the center of mass for thbounded by , y + 0 on the inter"al % Gi"edensity d-. is constant%

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Center of Mass Coordinat

)he coordinates of the center of mass, , are then

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Example !

*etermine the center of mass for theregion bounded by and %

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)ind center of mass of a thin +late of gien density

coering the following region-

and , constant density.

EAME 3

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Example " # $ensity is notconstant

4ind the center of mass of a thin late co"erregion bet!een the (a-is and the cur"e , 15the lates density at the oint -, y. is %

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)ind center of mass of a thin +late of gien density

the region, , !a!is, , density function,

0%6 1 1%6 2 2%6 3 3%6 7 7%6 6 6%60

0%6

1

1%6

2

2%6

Example %

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&'A *+,