ee101 l3 center of mass
TRANSCRIPT
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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 *+,