Friday, October 30, 1998

Chapter 8: Center of Gravity Moment of Inertia

x

xm gx m gx m gx m gx

m g m g m g m gCMN N

N

( ... )

( ... )1 1 2 2 3 3

1 2 3

xm x m x m x m x

m m m mCMN N

N

( ... )

( ... )1 1 2 2 3 3

1 2 3

Or, in shorthand notation xm x

mCM

i ii

N

ii

N

1

1

We can easily extendthis to 2-dimensionalobjects by finding acenter of mass in they-direction:

ym y

mCM

i ii

N

ii

N

1

1

What is the center of mass of this system:

(-1,0) m2 kg

(1,0) m1 kg

(0,1) m1 kg

xCM

( )( ) ( )( ) ( )( )

( ).

2 1 1 0 1 1

2 1 10 25

kg m kg m kg m

kg m

(-1,0) m2 kg

(1,0) m1 kg

(0,1) m1 kg

yCM

( )( ) ( )( ) ( )( )

( ).

2 0 1 1 1 0

2 1 10 25

kg m kg m kg m

kg m

CM

All the equations and laws we examinedin linear motion assumed point masses.

For extended (real) objects, these equationsreally describe the motion of the center ofmass of the objects.

x x v t a tCM CM CM CM 0 01

22

v v a tCM CM CM 0

The instantaneous velocity of a piece ofan extended object may not equal thevelocity of the center of mass.

For example, let’s look at the way thispillow flies across the room!

If I just asked you to plotthe horizontal velocity ofthe red square as a functionof time, what would such aplot look like?

v

vCM

t

Oscillates aroundthe center of massvelocity--sometimesfaster, sometimesslower.

Based on our definition of the center of mass,the velocity of the center of mass can beobtained if we know the velocities of all thelittle pieces of our system.

vm v

mCM

i ii

N

ii

N

1

1

Similarly for the acceleration of the centerof mass...

ama

mCM

i ii

N

ii

N

1

1

Clearly, the motion of the dumbells willbe quite different, even though the velocityof the center of mass is identical.

v v vCM1 2

vCM

v v vCM1 2

Ft

vCM

Ft

Ft

pinned totable

r

m

The tangential force results in a tangentialacceleration.

F mat t

Ft

pinned totable

r

m

It also creates a torque about the pinned point.

Fr ma rt t

Ft

pinned totable

r

m

Recalling that tangential acceleration is relatedto angular acceleration, we get

ma r m r r mrt ( ) 2

Ft

pinned totable

r

m

This expression is good so long as ourconnecting rod/string is massless.

mr 2