AREA
AND
MASS
MOMENT
OF
INERITA: the obtained resulting its resistance
in in elemental
areas
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sections in memberschange
and
in change
matter
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is of to
force
Fr² body, of
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of
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mass GYRATION
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radius application
and by conjunction
lamina
of (MO).
Thus
replaced and of the
study virtuethe of
the of
and inertia and
point
area moment the7.1. inertia
297
with measure OF
a moment
inertia
of its force multiplied
by moment
of
my2 by
inertia
of Fy2 is Ax2 is Fig.
made theorem statemoments identity
the body dealing in RADIUS laminaof
of inertia
of
CHAPTER 7
of = = used to moment
between
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having and x gives refence
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to axis
able
parallel inertia
moment Fx
×
Fx aboveplane of motion. is
while accordingly ANDany
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inertia
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:objectives
Learning
will
the moment
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ad the
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where mass
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yg J If cate 7.1.
to
, s
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7.2
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+
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dAy² axis.
and y-axis axis lamina
from axis centroidal
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anyaxís x A about +
about power its the ...(7.3)
x-X any from of Let distance aboutdAh? the
an parallel at
about has of between axisabout number locatedcomponent of
about aboutand x-axis.
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area fourth lamina an
distaDce its = -x
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the be +y)
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ENGINEERING dA
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is and of of dA the 0 ,
moment) inertia A? andI,, consists
= cummulative
axis.
from
that
G
of its gravity x-x component= of - z = = =
Theorem of
inertia
of to the components the inertia
x-axis. dA y LAA
OF
298ºA
TEXTBOOK I of
of
sum square + to
referenceof CG parallel of dA
first moment
second moment MOI lamina inertia h²
of the = the of
the
is
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the moment
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v (called
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by distance
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axes.
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where ofmm. The through AA
passing suchdistance Then Also
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is it.
, ...(7.7)moment
299 ..(76) dD
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AREA 7.4
Fig.
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oz oz dA
INERTIA lamina
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distance
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OF
an thetwo through the dA the DB
MOMENT i.e., and
about each are of passing y)
component of of G (a) is
to the
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= inertia
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standard
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intersecting (*2 y) 12 breadth 7.4
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r perpendi
thWhere cular
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in
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BD3
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section 12
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n te te of
AND
MASS
MOMENT
OF
INERITA: the obtained resulting its resistance
in in elemental
areas
dA
this and
composite
sections in memberschange
and
in change
matter
presented (E) the bodies.
is of to
force
Fr² body, of
anymeasure offers
aoflamina quantity
moment
force
of deformation
rigid dA)y
of force.
of resists
theorem product the body
all
shapes inertia of body. a of (y
of essentially
body rotation of
a mass that
of
gyration
subject axis action
of then moment
isthat
the thethe or =
different
of or resistance
-axis
perpendicular
and
product distance
x, second
or
the area of which deflection with
of
the about
a
point is
mass GYRATION
line
the inertia secondabout
radius application
and by conjunction
lamina
of (MO).
Thus
replaced and of the
study virtuethe of
the of
and inertia and
point
area moment the7.1. inertia
297
with measure OF
a moment
inertia
of its force multiplied
by moment
of
my2 by
inertia
of Fy2 is Ax2 is Fig.
made theorem statemoments identity
the body dealing in RADIUS laminaof
of inertia
of
CHAPTER 7
of = = used to moment
between
momentthe areabody denoteArea a
having and x gives refence
a
to axis
able
parallel inertia
moment Fx
×
Fx aboveplane of motion. is
while accordingly ANDany
principal the = further a property
respectively
Afterbe area/inJdertyfminieaassarea/mass
, force
=
moment
the
moment of
inertia applied
inertia
of With
INERTIA
=
I,,
of the
:objectives
Learning
will
the moment
(x)
distance
perpendicular
recalled is in
a
of uniform
of
(MO)
lamina.
reader prove terms of Fx as
moment
the inertia the moment
determ i
polnae
r Momentto of force
referred
F of
called moment m to is and
OF inertia
defineerplain
the Moment
thechapter,
ad the
be tormis of
andrefers
of and
or
velocity
MOMENTthe
may Moment
paramete r A
where mass
bending,
state this tin
iswhich
rest
Mass Inertia bending. of Comprising
MOment
It of The angular
yg J If cate 7.1.
to
, s
(h
be IX.
Fig.
7.1 will dA(2hy) axis
the A
area Fig.
7.2
is AA be the
y of will
dA) Lamina axis about
+
(y Area
A the AA AA
dAy² axis.
and y-axis axis lamina
from axis centroidal
B
x-axis
r-axis axis square of units about
anyaxís x A about +
about power its the ...(7.3)
x-X any from of Let distance aboutdAh? the
an parallel at
about has of between axisabout number locatedcomponent of
about aboutand x-axis.
MECHANICS inertia MOI product h lamina inertiais
area fourth lamina an
distaDce its = -x
dA section a MOI infinite
the be +y)
dA mm, about then y)² entire x
areaarea of of the distance about dA of Ah?
because
ENGINEERING dA
Er'
moment= product plane the the a to elemental
areaObviously+ dA(h moment =
'AjBB =
a are
in MOIand 7.2. laminaat parallel (h =
ofmoment
of an Ah?
of are
MOImeasurementsa the Fig.
is and of of dA the 0 ,
moment) inertia A? andI,, consists
= cummulative
axis.
from
that
G
of its gravity x-x component= of - z = = =
Theorem of
inertia
of to the components the inertia
x-axis. dA y LAA
OF
298ºA
TEXTBOOK I of
of
sum square + to
referenceof CG parallel of dA
first moment
second moment MOI lamina inertia h²
of the = the of
the
is
the axis of thecentre 'A A its elemental moment
the moment
to andWith is through
is fromof
elemental
v (called
dA ofthe of When Parallel
by distance
Obviously which The
Moment
That
gives:
Likewise: prescribedunits equalits (mass) where I
axes.
Proof: y Now,
where ofmm. The through AA
passing suchdistance Then Also
moment Thelength. small
7.1.1. is
axis areatwo axis one
is it.
, ...(7.7)moment
299 ..(76) dD
Elemental
area
dA Lamina L. L. -
its
= =
oy ox of
Fig.
73 (b) ratio below:
MASS axisaxis B
AND the the the a).
presented
about
about of 7.4
AREA 7.4
Fig.
2 root (Fig.
oz oz dA
INERTIA lamina
lamina d-x square
axis axis5 d
are depth 3
axissum axesat two through
lamina. The
distance
from + d,
otherit. about component
aboutx² the the sections - 12
OF
an thetwo through the dA the DB
MOMENT i.e., and
about each are of passing y)
component of of G (a) is
to the
equal plane 0z,
= inertia
inertia lamina
standard
passesoy and
0x
oy).from + b db3 b)
lamina about
intersecting (*2 y) 12 breadth 7.4
and
is theand dA + of of
Theorem
Axis
perpendi
plane
lamina axisox
culalamina
r perpendi
thWhere cular
7.3,
in
axes
dA
lamina elemental
area
= elemental
p2
(r moment
dA
moment
a concentrated
bechangefrom
given of of can ;
k= a
of
common
gyrationwith
(Fig.
BD3
bd
section 12
a
thetheand Fig.
inertia
of of of other
each
to reference
planeinertia
lying
theof
perpenmtvdail1nictuecomponent
theof
to axestonormal
axis
dA
=
the the tlrohasefrpeoictnt ion
= = =
of
mass)
Gyration
a
no about
point
radiusradius
I
inertia
Ak?
I=
of of
bd3 12
section
momthee7nNt uolamrents L 2 2 rectangular
of
to of With
of
el mentalinertiainertia
of -A A
M M
(or is
area
there
that
inertia
considered
to of
called
the between
of
moment
radius inertia
rectangular
J'rot ofMoment ofMoment of entirethat
Radius distance
suchfofmomentaxis7 relation
form
and the
thearea.of
moment
Apparently
its
hcllow
The an the Iskunapont thengivenoation. k in to a
For
the But, Z13. # ta the 2s, The cTationput inertiaIhe i
a
For
n te te of
