Rotational Inertia

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Rotational Inertia
Circular Motion

Objects in circular motion
have kinetic energy.
• K = ½ m v2
v

r
w
m
The velocity can be
converted to angular
quantities.
• K = ½ m (r w)2
• K = ½ (m r2) w2
Integrated Mass
Ki 
1
2
( m i )( w ri )
Ki 
1
2
w ( ri m i )

K rot 
1
2
The kinetic energy is due to
the kinetic energy of the
individual pieces.

The form is similar to linear
kinetic energy.
2
2
2
K rot 

1
2
Iw
w ( ri m i )  w
2
2
2
1
2
2
r
i
2
m
• KCM = ½ m v2
• Krot = ½ I w2
i

The term I is the moment of
inertia of a particle.
Moment of Inertia Defined

The moment of inertia measures the resistance to a
change in rotation.
• Mass measures resistance to change in velocity
• Moment of inertia I = mr2 for a single mass

The total moment of inertia is due to the sum of
masses at a distance from the axis of rotation.
N
I 

i 1
m i ri
2
Two Spheres

w
A spun baton has a moment
of inertia due to each
separate mass.
• I = mr2 + mr2 = 2mr2
m
r
m

If it spins around one end,
only the far mass counts.
• I = m(2r)2 = 4mr2
Mass at a Radius

Extended objects can be
treated as a sum of small
masses.

I 


The total moment of inertia is
A straight rod (M) is a set of
identical masses Dm.

( m i ) ri
2
Each mass element
contributes
mi  (M / L )Dr
I  ( M / L )  ri D r
2
distance r to r+Dr

length L
The sum becomes an
integral
I  (1 / 3 ) ML
axis
2
Rigid Body Rotation

The moments of inertia for many shapes can found
by integration.
• Ring or hollow cylinder: I = MR2
• Solid cylinder: I = (1/2) MR2
• Hollow sphere: I = (2/3) MR2
• Solid sphere: I = (2/5) MR2
Point and Ring

The point mass, ring and
hollow cylinder all have the
same moment of inertia.

• I = MR2

The rod and rectangular
plate also have the same
moment of inertia.
• I = (1/3) MR2
All the mass is equally far
away from the axis.

The distribution of mass from
the axis is the same.
M
R
M
R
M
M
length R
axis
length R
Parallel Axis Theorem

Some objects don’t rotate
about the axis at the center
of mass.

The moment of inertia for a
rod about its center of mass:
h = R/2

M
The moment of inertia
depends on the distance
between axes.
I  I CM  Mh
2
axis
(1 / 3 ) MR
2
 I CM  M ( R / 2 )
I CM  (1 / 3 ) MR
2
I CM  (1 / 12 ) MR
 (1 / 4 ) MR
2
2
2
Spinning Energy

How much energy is stored
in the spinning earth?

The earth spins about its
axis.
• The moment of inertia for a
sphere: I = 2/5 M R2
• The kinetic energy for the
earth: Krot = 1/5 M R2 w2
The energy is equivalent to
about 10,000 times the solar
energy received in one year.
• With values: K = 2.56 x 1029 J
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