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Physics IClass 13

GeneralRotational Motion

Rev. 19-Feb-06 GB

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Definitions

A n g u l a r P o s i t i o n :

( n o r m a l l y i n r a d i a n s )A n g u l a r D i s p l a c e m e n t : 0

A v e r a g e o r m e a n a n g u l a r v e l o c i t y i s d e f i n e d a s f o l l o w s :

ttt 0

0avg

I n s t a n t a n e o u s a n g u l a r v e l o c i t y o r j u s t “ a n g u l a r v e l o c i t y ” :

tdd

tlim

0t

W a i t a m i n u t e ! H o w c a n a n a n g l e h a v e a v e c t o r d i r e c t i o n ?

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Direction of Angular Displacement and Angular Velocity

•Use your right hand.

•Curl your fingers in the direction of the rotation.

•Out-stretched thumb points in the direction of the angular velocity.

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Angular Acceleration

A v e r a g e a n g u l a r a c c e l e r a t i o n i s d e f i n e d a s f o l l o w s :

ttt 0

0avg

I n s t a n t a n e o u s a n g u l a r a c c e l e r a t i o n o r j u s t “ a n g u l a r a c c e l e r a t i o n ” :

2

2

0t tdd

tdd

tlim

T h e e a s i e s t w a y t o g e t t h e d i r e c t i o n o f t h e a n g u l a r a c c e l e r a t i o n i st o d e t e r m i n e t h e d i r e c t i o n o f t h e a n g u l a r v e l o c i t y a n d t h e n … I f t h e o b j e c t i s s p e e d i n g u p , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n

a r e i n t h e s a m e d i r e c t i o n . I f t h e o b j e c t i s s l o w i n g d o w n , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n

a r e i n o p p o s i t e d i r e c t i o n s .

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Equations for Constant

1 . 00 tt

2 . 202

1000 )tt()tt(

3 . )tt)(( 0021

0

4 . 202

100 )tt()tt(

5 . 020

2 2

xva

We recommend always using radians for angles.

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Relationships AmongLinear and Angular Variables

MUST express angles in radians.rs rv

ra tangential

rr

rr

va 2222

lcentripeta

T he rad ia l d irec tion is defined to b e +outw ard fro m the center.

lcentripetaradial aa

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Energy in Rotation

C o n s i d e r t h e k i n e t i c e n e r g y i n a r o t a t i n g o b j e c t . T h e c e n t e r o fm a s s o f t h e o b j e c t i s n o t m o v i n g , b u t e a c h p a r t i c l e ( a t o m ) i n t h eo b j e c t i s m o v i n g a t t h e s a m e a n g u l a r v e l o c i t y ( ) .

2ii

2212

i2

i212

ii21 rmrmvmK

T h e s u m m a t i o n i n t h e f i n a l e x p r e s s i o n o c c u r s o f t e n w h e na n a l y z i n g r o t a t i o n a l m o t i o n . I t i s c a l l e d r o t a t i o n a l i n e r t i a o rt h e m o m e n t o f i n e r t i a .

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Rotational Inertiaor Moment of Inertia

F o r a s y s t e m o f d i s c r e t e “ p o i n t ” o b j e c t s :2

ii rmI

F o r a s o l i d o b j e c t , u s e a n i n t e g r a l w h e r e i s t h e d e n s i t y :

dzdydxrI 2

W e m a y a s k y o u t o c a l c u l a t e t h e r o t a t i o n a l i n e r t i a f o r p o i n t o b j e c t s , b u t w e w i l lg i v e y o u a f o r m u l a f o r a s o l i d o b j e c t o r j u s t g i v e y o u i t s r o t a t i o n a l i n e r t i a .

I for a solid sphere: 252 RMI

I for a spherical shell: 232 RMI

R

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Characteristics ofRotational Inertia

T h e ro ta tio n a l in e rtia o f a n o b jec t d ep en d s o n I ts m ass . I ts sh a p e . T h e ax is o f ro ta tio n . N O T th e a n g u la r v e lo c ity o r a cc e le ra tio n .

T h e ro ta tio n a l in e rtia is a m ea su re o f h o w d iffic u lt it is to g e t a no b jec t to s ta r t ro ta tin g o r to s lo w d o w n o n c e s ta r te d .

F o r tw o o r m o re o b je c ts ro ta tin g a ro u n d a co m m o n a x is , th e to ta lro ta tio n a l in e r tia is th e su m o f e ach in d iv id u a l ro ta tio n a l in e r tia .

iII

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Introduction to Torque

For linear motion, we have “F = m a”. For rotation, we have

IThe symbol “” is torque. We will define it more precisely nexttime.

Torque and angular acceleration are alwaysin the same direction in Physics 1 becausewe consider rotations about a fixed axis.

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Correspondence BetweenLinear and Rotational Motion

xvaIm F

221 IK I

You will solve many rotation problemsusing exactly the same techniques youlearned for linear motion problems.

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Class #13Take-Away Concepts

1 . D e f i n i t io n s o f r o t a t io n a l q u a n t i t i e s : , , .2 . C e n t r ip e ta l a n d t a n g e n t i a l a c c e le r a t io n .

3 . R o ta t io n a l i n e r t i a : 2ii rmI

4 . R o ta t io n a l k in e t i c e n e r g y : 221 IK

5 . I n t r o d u c t io n to to r q u e : I6 . C o r r e s p o n d e n c e

x v aIm F

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Class #13Problems of the Day

___1. Lance Armstrong is riding his bicycle due east in theTour de France bicycle race. What is the direction ofthe angular velocity of his bicycle wheels?

A) North.B) South.C) East.D) West.E) Up.F) Down.

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Class #13Problems of the Day

2. A carousel takes 8 seconds to make one revolutionwhen turning at full speed. It takes exactly one revolutionto reach its full speed from rest. What is the magnitude ofits angular acceleration while speeding up, assuming thatits angular acceleration is constant?

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Activity #13 Introduction to Rotation

Objective of the Activity:

1. Think about rotation concepts.2. Try changing the rotational inertia of a simple

object and see how that affects I .