Physic Insteresting

8/9/2019 Physic Insteresting

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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

R
http://www.youtube.com/watch?v=V3UsrfHa4MQhttp://www.youtube.com/watch?v=V3UsrfHa4MQ

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Physic Insteresting