What Is Rotational Inertia?!
Examples!
KE total = ! KE i
ri
2%
"1
=! $ m i ( ri( ) '
#2
&
(! m r )
2
i i
is clearly an expression designed for discrete masses
(example: a mass at each end of a rod), so let’s try it in that context. !
2
i i
vi = ri!
"1
%
=! $ m i v i 2 '
#2
&
1
=
2
!m r
1.) Determine the moment of inertia about the central axis for the
set-up shown below. Assume the rod is massless and the masses
equal in magnitude. !
central axis!
!
m!
(2
2
The ! m i ri term is called the
body’s “moment of inertia” about
the central axis.!
m!
L
2
!
2
1
" L%
" L%
Icm = ! m i ri 2 = m $ ' + m $ ' = mL2
# 2&
# 2&
2
1.)!
Examples!
!m r
3.)!
Examples!
2.) Determine the moment of inertia about an axis through one of
the masses. Again, assume the rod is massless. !
is clearly an expression designed for discrete masses
(example: a mass at each end of a rod), so let’s try it in that context. !
2
i i
axis of
interest
1.) Determine the moment of inertia about the central axis for the
set-up shown below. Assume the rod is massless and the masses
equal in magnitude. !
central axis!
m!
m!
m!
L
!
m!
L
!
!
2.)!
4.)!
Examples!
Examples!
2.) Determine the moment of inertia about an axis through one of
the masses. Again, assume the rod is massless. !
axis of
interest
3.) Determine the moment of inertia about an axis a length L units
to the left of the left mass. Again, assume the rod is massless. !
axis of
interest
!
m!
!
m!
L
m!
!
L
!
L
!
Ioutside = ! m i ri 2 = m ( L ) + m ( 2L ) = 5mL2
2
Iend = ! m i ri 2 = m ( 0 ) + m ( L ) = mL2
2
m!
2
2
Parting shot: The moment of inertia gets bigger and bigger as you get
farther and farther away from the body’s center of mass.!
5.)!
mass
)(differential volume)
volume
" M %
= $
( 2!r t dr )
# !R 2 t '&
dm = (
Examples!
3.) Determine the moment of inertia about an axis a length L units
to the left of the left mass. Again, assume the rod is massless. !
axis of
interest
7.)!
!
" M%
= $ 2'
#R &
( 2 r dr )
" 2M %
= $ 2'
#R &
( r dr )
Example of derivation of moment of
inertia for a continuous mass--NOT
something you will be tested on!!
dr!
So the moment of inertia is:!
m!
L
!
m!
L
I=
!
r!
r 2 dm
" 2M %
= ! r $ 2 'r dr
r=0
#R &
R
!
2
=
2M R 3
r dr
R 2 !r=0
=
2M ( r 4 +
R 2 )* 4 ,-
(
6.)!
R
r=0
M
=
R4 . 0
2R 2
1
= MR 2
2
)
Disk of mass “m”, radius “R” and
thickness “t.”!
8.)!
Bottom line: Every extended object has a moment of inertia expression
that allows you to determine how much rotational inertia the body has if
spun about that axis.!
8.)!
PARALLEL AXIS THEOREM!
The Parallel Axis Theorem simply states: If you have a allows you to determine the
moment of inertia about !
8.)!