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Equation of Rotational Motion

X

Y

prir

iP

r r

X

Y

prir

( )p i i i i im M r F r f r a

( )i p a a α r ω ω r

( ) [ ( ) ( ( ))]p i i pm M r a r α r r ω ω r

( ) {( ) [( ) ( ) ]

( ) [ ( )]}p i i p x p yM m x y a a

x y x y

k i j i j

i j k i j

2 2

2

( ) [ ( ) ( ) ]

( ) [ ( ) ( ) ]p i i p x p y

p i i p x p y

M m y a x a x y

M m y a x a r

k k

Kinetics of a Rigid Body: Force and Acceleration

Chungnam National University

Equation of Rotational Motion

X

Y

prir

2

2

2

2

( ) [ ( ) ( ) ]

( ) [ ( ) ( ) ]

( ) [ ( ) ( ) ]

( )( ) ( )( ) ( )

p i i p x p y

p i p x p y

p i p x p y

p x p y

M m y a x a r

M dm y a x a r

M dm y a x a r

ydm a xdm a r dm

( ) ( )

G i

G G

m dm

mx my xdm ydm

mx my

r r

i j i j

i j

m dm

( ) ( )p p x p y pM ym a xm a I 2( )pI r dm where If point P = point G G GM I

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Equation of General Motion

G GM I

( ) ( )p p x p y pM ym a xm a I 2( )pI r dm where

If point P = point G

The sum of the moments of all the external forces computed about the body’s mass center G is equal to the product of the moment of inertia of the body about an axis passing through G and the body’s angular acceleration.

Chungnam National University

Equation of General Motion

( ) ( )p p x p y pM ym a xm a I 2 2 2( )p G GI I mr I m x y

2 2

2 2

( ) ( ) ( )

[ ( ) ] [( ) ]

[ ( ) ] [( ) ]

( ) ( )

p p x p y G

p x p y G

G x G y G

G x G y G

M ym a xm a I m x y

ym a y xm a x I

ym a x xm a y I

ym a xm a I

( ) ( ) ( )p G x G y G k pM ym a xm a I M 2

G P a a r r

The moments of the external forces about P are equivalent to the sum of the “kinetic moments” of the components of about P plus the “kinetic moment” of .

Gma

GI

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General Application of the Equations of Motion

G GM I

( )p k pM M

( )

( )x G x

y G y

F m a

F m a

or

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Equations of Motion: Translation

0GM ( )

( )x G x

y G y

F m a

F m a

Rectilinear Translation

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Equations of Motion: Translation

Chungnam National University

Equations of Motion: Translation

Chungnam National University

( )

( )n G n

t G t

F m a

F m a

Equations of Motion: Translation

Curvilinear Translation

0GM

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Equations of Motion: Curvilinear Translation

Chungnam National University

Equations of Motion: Curvilinear Translation

Chungnam National University

2( )

( )n G n G

t G t G

F m a m r

F m a m r

Equations of Motion: Rotation about a Fixed Axis

G GM I

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2( )

( )n G n G

t G t G

F m a m r

F m a m r

Equations of Motion: Rotation about a Fixed Axis

G GM I Moment equation can be replaced by arbitrary point P on or off the body provided one accounts for the moments produced by

about the point.

Do we know this force?

In order to eliminate the unknown force , it is convenient to sum moments about the pin point at O.

( )p k pM M

( )k pM ( ) , ( ) ,G n G t Gm a m a I

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2( )

( )n G n G

t G t G

F m a m r

F m a m r

Equations of Motion: Rotation about a Fixed Axis

In order to eliminate the unknown force , it is convenient to sum moments about the pin point at O.

2

( ) ;

( ) ( )

( )

o k o

o G G t G G G G

G G o

M M

M r m a I r m r I

I mr I

2( )o G G oM I mr I

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Equations of Motion: Rotation about a Fixed Axis

Chungnam National University

Equations of Motion: Rotation about a Fixed Axis

Chungnam National University

Equations of Motion: Rotation about a Fixed Axis

Chungnam National University

Chungnam National University

Equations of Motion: General Plane Motion

Freebody diagram Kinetic diagram

( )

( )x G x

y G y

F m a

F m a

G GM I

( )p k pM M or

Chungnam National University

Chungnam National University

Chungnam National University

Chungnam National University

Chungnam National University

Chungnam National University

Chungnam National University

Chungnam National University