Finding Moments of Inertia
2.003J/1.053J
Dynamics and Control I, Spring 2007
Professor Thomas Peacock
3/14/2007
Lecture 11
2D Motion of Rigid Bodies: Finding Moments of Inertia, Rolling Cylinder with Hole Example
1
Finding Moments of Inertia
Figure 1: Rigid Body.
Figure by MIT OCW.
I
C
�
= m i
| ρ i
| 2 i
�
= m i
( x 2 i
+ y 2 i
) i
I
C is the Moment of Inertia about C.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Finding Moments of Inertia
Example: Uniform Thin Rod of Length L and Mass M
2
Figure 2: Uniform thin rod of length L and mass M .
Figure by MIT OCW.
I
C
�
= m i
( x 2 i
+ y 2 i
)For very thin rod, y i is small enough to neglect.
i
�
≈ m i x
2 i i
Rod has mass/length = ρ .
Convert to integral.
I
C
=
� rod x 2 dm dm = ρdx
I
C
�
L/ 2
= x 2 ρdx
�
−
L/ 2
= ρ x
3
3 �
L/ 2
−
L/ 2
= ρ
L 3
12
We know that mass M = ρL .
Therefore:
I
C
=
M L 2
12
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Example: Rolling Cyinder with a Hole
Example: Uniform Thin Disc of Radius R
3
Figure 3: Uniform thin disc of radius R .
Figure by MIT OCW.
Let ρ = mass/area.
Consider a sliver that is a distance r from the center on this disc of radius R .
I
C
�
= m i
( x
2 i
2
+ y i
)
� i
= ( x 2 + y 2 ) dm
=
� disc
R r
2
2 πrρdr
=
�
0
R
ρ 2 πr 3
0 dr = 2 πρ
R 4
4
= πR 2 ρ
R
2
2
Mass of Disc: M = πR 2 ρ .
Thus,
I
C
=
M R 2
2
Example: Rolling Cyinder with a Hole
Find the equation of motion for a cylinder with a hole rolling without slip on a horizontal surface.
In the hole with center A, R
2
= R/ 2.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Example: Rolling Cyinder with a Hole 4
Figure 4: Rolling cylinder with hole shown at 2 distinct positions.
Figure by
MIT OCW.
Kinematics
2 Constraints: 1.
Rolling on surface 2.
No slip condition
Use 1 generalized coordinate θ to describe the motion
Only need 1 equation.
For this example, we will use the work-energy principle to obtain the equation.
1.
Gravity is a potential force.
2.
Normal force on object: At point of contact, velocity is zero so no work done.
No work done by external forces therefore T + V = constant.
T = 1
2
M | v
C
| 2 + 1
2
I
C
| ω | 2 .
Need center of mass.
Where is the center of mass?
Below O, because of hole.
Kinetics
Center of Mass Calculation
First find position of center of mass.
We know the center of mass of disc without hole: Point O.
Can think of the hole to be “negative mass.”
�
Consider moments about OX at point O
ρπR 2 (0) = ρ ( πR 2 −
πR 2
) OC − ρ
4
πR 2
4
R
2
Distance from O to O is zero.
ρ ( πR 2 − πR
2
4
) OC : Mass moment of cylinder with the hole.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Example: Rolling Cyinder with a Hole 5
ρ
πR 2 R
4 2
: Mass moment of the hole.
3 π
4
( OC ) =
πR
8
⇒ OC =
R
6
Calculation of 1
2
M | v
C
| 2
We know what r
C is.
r
C is from point B to point C.
x
C
= Rθ −
R
6 sin θ y
C
= R −
R
6 cos θ
Differentiate: x
C
= Rθ −
R
6
θ
˙ cos θ y ˙
C
=
R
6
θ
˙ sin θ
1
2
M v
2
C
1 �
=
2
πR
=
1 3
2 4
πR
2
2
ρ −
�
ρ R
π
R
4
2
2 θ
˙ 2
ρ
+
�
( ˙
R 2
36
2
C
θ 2
2 y
C
) cos 2 θ −
R 2 θ
˙ 2
3 cos θ +
R 2 θ 2
36 sin 2 θ
�
1
2
M v 2 c
=
1 3
2 4
πR 4 θ
˙ 2
�
ρ 1 −
1
3 cos θ +
1
36
�
Calculation of 1
2
I
C
| ω | 2
2nd term of kinetic energy is 1
2
I
C
| ω | 2 .
What is I
C
?
We want to find I cwh
C
.
cwh: cylinder with hole mc: missing cylinder cyl: cylinder
First find I cwh
O around O .
Then shift to C with the Parallel-Axis Theorem.
I cyl
O
= I cwh
O
+ I mc
O
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Example: Rolling Cyinder with a Hole 6
I cwh
O
= I cyl
O
− I mc
O
I cyl
O
=
1
2
ρπR 2 R 2
I mc
O
= I mc
A
=
1
2
�
+ M
ρπ
R mc
2 �
( OA ) 2
R 2
4 4
+
�
← Parallel Axis Theorem
ρπ
R 2 � R 2
4 4
=
3
ρπR
32
4
I cwh
O
=
1
ρπR
2
4 −
3
ρπR
32
4 =
13
ρπR
32
4
Now:
I cwh
O
= I cwh
C
+ M cwh
( OC ) 2 ← Parallel Axis Theorem
I cwh
C
=
13
32
I cwh
C
ρπR
4
= I cwh
O
�
− ρπR
− M cwh
( OC ) 2
2
− ρ
πR 2
4
� R 2
36
=
37
96
ρπR
4
So we have that:
1
2
I
C
| ω | 2 =
1 37
2 96
ρπR 4 θ
˙ 2
T + V =
3
8
ρπR 4 θ
˙ 2
� 37
36
−
3
1 � cos θ +
37
192
ρπR 4 θ
˙ 2 + V = ρπR 4 θ
˙
2
� 37
−
1
64 8
� cos θ + V
Calculation of Potential Energy (V)
What is V ?
V = mgh =
3
πR
4
2
�
ρ R −
R cos θ
6
�
Finding Equation of Motion
So we have T + V = Constant d dt
( T + V ) = 0
Differentiate:
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].

Example: Rolling Cyinder with a Hole 7
2 ρπR 4 θ
˙
θ
� 37
64
−
1
8
� cos θ +
1
8
ρπR 4 θ
˙ 2 sin θθ
˙ +
1
8
πR 3 ρg sin θθ
˙ = 0
Equation of motion: motion is complicated.
Alternative Approach: Using Angular Momentum (Sketch)
Angular momentum about moving point B .
B is not on cylinder.
B is not on ground.
B is contact point between ground and cylinder.
τ
H
B
: Torque due to gravity
�
= H + r × P
B v
B
C C
: Moving Point ( Rθ
P = M v
C
τ
B
= d dt
H
B
+ v
B
× P
1.
Still have to find velocity and location of center of mass.
2.
Still have to find I
C
.
3.
But, even more work because need to take torques.
Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J
Dynamics and
Control I, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].