Transcript Moment of Inertia Lecture

Moment of Inertia ( I )
The property of an object that serves
as a resistance to angular motion.
Chapter 7 in text
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Moment of Inertia ( I )  Mass
• Moment of inertia is the angular
equivalent of mass.
• Moment of inertia is affected by both the
mass and how the mass is distributed
relative to the axis of rotation.
• Unlike mass which remains constant
regardless of the direction of motion, the
moment of inertia of an object changes
depending on the axis of rotation.
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Mathematically Defining I
• An object can be thought to be composed of
many particles of mass. Hence,
Ia 
N
2
 mi ri
i 1
Ia = moment of inertia about axis a
mi = mass of particle i
ri = radius from particle i to the axis of rotation
Each particle provides some resistance to
change in angular motion.
• The units are mass * squared length: kg*m2
•
•
•
•
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Calculate I for the baseball
N
bat using I   m r 2
a
1
a
i 1
1 kg
i i
2 kg
0.4 m
0.8 m
2
1 kg
0.4 m
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2 kg
b
0.4 m
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Solutions
Ia 
1
N
2
 mi ri
i 1
Ia = (1 kg)(0.4 m)2 + (2 kg)(0.8 m)2
Ia = 1.44 kg*m2
2
Ib = (1 kg)(0.4 m)2 + (2 kg)(0.4 m)2
Ib = 0.48 kg*m2
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Radius of Gyration (k)
• The distance from the axis of rotation to a point
where all of the mass can be concentrated to
yield the same resistance to angular motion.
• An averaging out of the radii (ri) of all the mass
particles. This allows all the mass to be
represented by a single radius (k).
Ia 
2
mka
• The distribution of an object’s mass has a much
greater affect on the moment of inertia than
mass.
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Moments of Inertia about 3 Axes of a Block
1
2
Axis 1
3
Axis 2
rc
ra
ra
rb
Axis 3
rc
rb
Applying, I a  mka , the axis with the
greatest radius of gyration (k) will have the
greatest moment of inertia because the mass
of the block doesn’t change.
2
Rotating about Axis 1, the distribution of the
block’s mass has the greatest average radius (k).
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3 Principal Axes for any Object
Maximum Moment of Inertia Axis (Imax)
Axis that has the largest moment of inertia
Minimum Moment of Inertia Axis (Imin)
Axis that has the smallest moment of inertia
Intermediate Moment of Inertia Axis (Iint)
Has an intermediate moment of inertia. Determined
not by its moment of inertia value, but rather because
it is perpendicular to the both Imax and Imin.
Note: All three axes are perpendicular to each other
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3 Principal Axes for a Human
in Anatomical Position
Longitudinal
Axis
Frontal
= Imax
(Cartwheel)
Transverse Axis
Transverse
(Back flip)
= Iint
Longitudinal = Imin
(Discus throw)
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Moments of inertia in kgm2
19
12
1
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2
4
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Golf Club Heads
Clubhead 1
Clubhead 2
Top View
If both clubheads have the same mass, which one has
the greatest moment of inertia in the plane shown?
Clubhead 2 has a greater distribution of mass, therefore
a greater moment of inertia and a greater resistance to
angular motion.
Perimeter weighted clubs are more forgiving on off
center hits.
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Cavity Back Putters
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Cavity Back Irons
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Muscle Back Irons
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The Modern Tennis Racquet
Early Version
Twisting
on off
center Hits
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Transition
Modern Version
Larger moment of
inertia reduces
twisting from off
center hits
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Force
and
F = ma and
Torque
 = I
• Newton’s Laws also apply to angular
motion.
• For every linear term, there is an
equivalent angular term.
• For example, torque is the angular effect of
force. Just like a net force produces an
acceleration resisted by the mass, a net
torque produces an angular acceleration
resisted by the moment of inertia.
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Linear Impulse and Angular Impulse
Ft = mv and

 t = I
• A net force acting for a period of time
produces a linear impulse that results in a
change in linear momentum.
• Likewise, a net torque acting for a period
of time produces an angular impulse that
results in a change in angular momentum.
• Where angular momentum is the product
of the moment of inertia and angular
velocity.
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Comparison of Linear and
Angular Quantities
LINEAR
ANGULAR
Time (t)
Linear Displacement (D)
Angular Displacement ()
Linear Velocity (v)
Linear Acceleration (a)
Force (F)
Mass (m)
Linear Momentum (mv)
Angular Velocity ()
Angular Acceleration ()
Linear Impulse (Ft)
Angular Impulse ( t)
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
Torque ( )
Moment of Inertia (I)
Angular Momentum (I)

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