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Rotational inertia symbol I
• Rotational inertia is a parameter that is
used to uantify how much torue it ta!es
to get a particular ob"ect rotating
• it depends not only on the mass of the
ob"ect# but where the mass is relative to
the hinge or a$is of rotation
• the rotational inertia is bigger# if more
mass is located farther from the a$is%
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Rotational inertia and torue
• To start an object spinning, a
torque must be applied to it
• The amount of torque requireddepends on the rotationalinertia (I) of the object
• The rotational inertia (I)depends on the mass of theobject, its shape, and on howthe mass is distributed
• Solid disk I ! " # R$
• The higher the rotation inertia,the more torque that is requiredto make an object spin
W= mg
T
&orue ' & ⋅ R
R
M
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rotational inertia e$amples
ob"ects have identical mass and length
arge rotational inertia
*mall rotational inertia
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+ow fast does it spin?
• ,or spinning or rotational motion# therotational inertia of an ob"ect plays thesame role as ordinary mass for simple
motion• ,or a given amount of torue applied to an
ob"ect# its rotational inertia determines its
rotational acceleration
the smaller therotational inertia# the bigger the rotationalacceleration
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-ig rotational
inertia
*mall rotational
inertia
Same torque,
different
rotational inertia
spins
slow
spins
fast
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Rolling down the incline
Which one
reaches the
bottom first# the
solid dis! or the
hoop? &heyhave the same
mass and
diameter%
The solid disk gets to the bottom faster because
it has the smaller rotational inertia
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*peed of rotation
• ,or motion in a straight line we tell how fast yougo by the velocity meters per second# miles perhour# etc%
• +ow do we indicate how fast something rotates?
• We use a parameter called rotational velocity#(symbol. /mega
Ω) simply the number ofrevolutions per minute for e$ample .. the numberof times something spins say in a second or
minute (rpm’s. revs per min)• for e$ample the rotational speed of the earth
spinning on it a$is is 0 revolution per day or 0revolution per 1 hours%
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/rdinary (linear) speed and
rotational speed
• the rod is rotatingaround the circle inthe countercloc!wise
direction• 2 points on the rod
have the SM! rotational speed
• &he red point in themiddle has only halfthe linear speed asthe blue point on the
end%every point on the line moves
through the same angle
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Ice %apades
*!aters farther from center must s!ate faster
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+urricanes
Most dangerous winds are at edges of hurricane
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Conservation of linear momentum
• 3f an ob"ect is moving with velocity v# it has linear
momentum: p ' m v
• 3f no outside forces disturb the ob"ect# it its linear
momentum is conserved• 3f ob"ects interact (e%g%# collide) the forces are
eual and opposite and cancel each other so the
linear momentum of the pair is conserved%
(p 2 4 p-)before ' (p 2 4 p-)after
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Rotational (angular) momentum J
• A spinning object has rotational momentum
symbolJ
• rotational momentum (J) =
rotational inertia (I) x rotational velocity (Ω)
• J = I Ω
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Conservation of rotational momentum
• 3f no outside torues disturb a spinningob"ect# it rotational momentum is conserved
• &he rotating masses on the rod !eep
spinning until the friction in the bearingslows it down% Without friction# it would !eepspinning%
•5ote that the total linear momentum is 6ero so thatdoes not come into play%
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Rotational momentum
• & ! rotational inertia (I)×
angular 'elocity (Ω
)• since the rotational momentum cant change
then if the rotational inertia changes, therotational 'elocity must also change to keep
the rotational momentum constant• r, I* Ω* ! I$ Ω2
• If the rotational inertia increases, then the
rotational 'elocity must decrease• if the rotational inertia decreases, then the
rotational 'elocity must increases
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Rotational momentum
demonstrations
• spinning ice s!ater
• divers
• +obermann sphere• bicycle wheel
• top
• gyroscope
/b"ects that have rotational momentum (*735) tend
not to loose it easily
-icycles
839:/
Ω
Ω
I*
I$
%onser'ation of J:I$ + I* Ω2 > Ω1
http://www.youtube.com/watch?v=AQLtcEAG9v0http://www.youtube.com/watch?v=AQLtcEAG9v0
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-ig rotational
inertia
small rotational
inertia
;ou can change your rotational inertia
I Ω = I Ω
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*pinning faster or slower
• When your arms are e$tended you have abig rotational inertia
• When you pull your arms in you ma!e
your rotational inertia smaller • 3f you were spinning with your arms out#
when you pull your arms in you will spin
faster to !eep your rotational momentumconstant
• &his wor!s in figure s!ating and diving
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:$ample
• A figure sater has a rotational inertia I! "hen herarms are stretche# out$ an# I% "hen her arms are
pulle# in close to her bo#y& If her angular velocity
is Ω! "hen she spins "ith her arms stretche# out$
"hat is her angular velocity "hen she pulls hersarms in$ so that I% = ' I! = & I!
• Solution: Angular momentum is conserve#$ so
I! Ω! = I% Ω%
since I!*I% = ! * & = %$ Ω% = % Ω!
+he #oubles her angular spee#&
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*pinning wheel defies gravity<
spinning
wheel
"yroscope. an ob"ect that canspin and rotate about three a$es
/nce it starts spinning
its a$le wants to !eep
spinning in the same
direction% 3t resists forces
that try to change the
direction of its spin a$is%
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9ivers use rotational momentum
conservation to spin• the diver starts spinning
when she "umps off the
board
• when she pulls her armsand legs in she ma!es her
rotational inertia smaller
• this ma!es her spin even
faster<
• +er C= follows the same
path as a pro"ectile
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,atural effects #ue to conservation of J
• -he length of the #ay is #etermine# by the time ittaes the .arth to complete one full spin about its
axis& /ig earth0uaes can alter the #istribution of
mass in the earth*s crust& -he #istribution of mass
#etermines the rotational inertia of the earth&• -he 1oon is getting farther from the .arth because
the .arth2s #aily rotation is slo"ing #o"n #ue to
friction of the ocean "aters on the ocean bottom& -he
#ecrease in the angular momentum of the .arth is
accompanie# by an increase in the angular
momentum of the 1oon in its orbit aroun# the .arth&
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&ornadoes
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9on’t fall off the stool<
http>www%youtube%comwatch?v'8@Asrf+a1MB
R
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http://www.youtube.com/watch?v=V3UsrfHa4MQhttp://www.youtube.com/watch?v=V3UsrfHa4MQ