Rotation of rigid bodies
A rigid body is a system where internal forces hold each part in the
same relative position
Kinetic Energy and Rotation. Moment of Inertia.
K rot   m v 
2
i i
1
2
i
1
2
2 2
1


m

r

m
r
 i i 2 i i 
2
i
K rot  I
1
2
i
I   mi ri 2
2
i
2
K  12 MvCM
 12 I CM  2
If
vCM

R
M   mi
i
(rolling without slipping) and
2
K  Mv
1
2
2
CM
I   r 2 dm
M   dm
I  kMR2
 vCM 
2
1
 kMR 

Mv

CM (1  k )
2
 R 
1
2
2
then
Note: I=kMR2, where k  1
Example: Two spheres have the same radius and equal masses. One is
made of solid aluminum and the other is a hollow shell of gold. Which
one has the biggest moment of inertia about an axis through its center?
Solid
aluminum
A. Solid Al
Mass is further
away from the axis
Hollow
gold
B. Hollow Au
C. Both the same
Example: Moment of inertia of a square of side L made with
four identical particles of mass m and four massless rods.
Axis
Axis
m
m
L
m
Axis
I  2mL2
m
m
m
m
m
L
m
L
m
m
I  mL2
m
I  2mL2
The moment of inertia depends on the position and orientation of the axis
Example: Three identical balls are connected with three identical,
rigid, massless rods. The moments of inertia about axes 1, 2 and 3
are I1, I2 and I3. Which of the following is true?
A. I1 > I2 > I3
B. I1 > I3 > I2
m
1
2
C. I2 > I1 > I3
I1 = m(2L)2 + m(2L)2 = 8mL2
L
I2 = mL2 + mL2 + mL2 = 3mL2
I3 = m(2L)2 = 4mL2
3
Example: Uniform rod of length L and mass M for rotations about the
perpendicular axis through its center.
x dx
dm  dx
L2
M
Density:  
L
3
3

1
L
L
L3 M L3
1
  
 
2
2
I   x dm   x dx           

 ML2
3  2   2  
12 L 12 12
L 2
Example: Heavy (real) pulleys. Two blocks of masses m1 and m2 (> m1 )
are connected through a string that goes through two different pulleys.
In case 1, the pulley is made of plastic. In case 2, the pulley is made of
iron. In both cases, mass m1 is initially at rest on the floor and mass m2
hangs at distance h from the floor. Both systems are released
simultaneously. In which case does mass m2 hit the floor first?
R

I cilinder  12 MR 2
m2
m1
Case 1; Case 2; Same for both
v
h
K  12 I 2  14 MR 2 2  14 Mv 2
v
h
m2 gh  m1 gh  m1v  m2 v  Mv
1
2
2
1
2
2
1
4
If no slipping:
v = R
2
2  m2  m1  gh
v 
M
m1  m2 
2
Parallel-axis theorem



I   mi ri  mi ri ,cm  d
2
  m r
2
i
2
i ,cm

 
 2ri ,cm d  d 2  mi ri 2,cm   mi d 2
I  I cm  Md
2
EXAMPLE: Rod of mass M and length L about the axis through one end:
I axis CM
Axis A
d = L/2
Axis through CM
Iaxis A  Iaxis CM  Md 2
1
 ML2
12
1
 L   1 ML2
2

ML  M  
12
3
2
2