Transcript Section 10.9 - Federal University of Rio de Janeiro

MASS MOMENT OF INERTIA (Section 10.9)
Today’s Objectives:
Students will be able to :
a) Explain the concept of the mass
moment of inertia (MMI).
b) Determine the MMI of a
In-Class Activities:
composite body.
• Check homework, if any
• Reading quiz
• Applications
• MMI: concept and definition
• Determining the MMI
• Concept quiz
• Group problem solving
• Attention quiz
READING QUIZ
1. The formula definition of the mass moment of inertia about
an axis is ___________ .
A)  r dm
B)  r2 dm
C)  m dr
D)  m2 dr
2. The parallel-axis theorem can be applied to determine
________ .
A) only the MoI
B) only the MMI
C) both the MoI and MMI
D) None of the above.
Note: MoI is the moment of inertia of an area and MMI is the mass
moment inertia of a body
APPLICATIONS
The large flywheel in the
picture is connected to a large
metal cutter. The flywheel is
used to provide a uniform
motion to the cutting blade.
What property of the flywheel is
most important for this use?
How can we determine a value
for this property?
Why is most of the mass of the
flywheel located near the
flywheel’s circumference?
APPLICATIONS (continued)
If a torque M is applied to a
fan blade which is initially
at rest, its angular speed
begins to increase.
On which property (P) of
the fan blade does the
angular acceleration ()
depend? How can we
determine a value for P?
What is the relationship
between M, P, and ?
CONCEPT OF THE MMI
T
G
·
Consider a rigid body with a center of
mass at G. It is free to rotate about the z
axis, which passes through G. Now, if we
apply a torque T about the z axis to the
body, the body begins to rotate with an
angular acceleration .
T and  are related by the equation T = I  . In this equation, I
is the mass moment of inertia (MMI) about the z axis.
The MMI of a body is a property that measures the resistance
of the body to angular acceleration. This is similar to the role
of mass in the equation F = m a. The MMI is often used
when analyzing rotational motion (done in dynamics).
DEFINITION OF THE MMI
p
Consider a rigid body and the
arbitrary axis p shown in the figure.
The MMI about the p axis is defined
as I = m r2 dm, where r, the “moment
arm,” is the perpendicular distance
from the axis to the arbitrary element
dm.
The MMI is always a positive
quantity and has a unit of kg ·m2
or slug · ft2.
RELATED CONCEPTS
Parallel-Axis Theorem:
Just as with the MoI for an area, the
parallel-axis theorem can be used to
find the MMI about a parallel axis p’
that is a distance d from the axis
m
through the body’s center of mass G.
The formula is Ip’ = IG + (m) (d)2
(where m is the mass of the body).
d
G·
The radius of gyration is similarly defined as
k = (I / m)
Finally, the MMI can be obtained by integration or by the
method for composite bodies. The latter method is easier
for many practical shapes.
p’
EXAMPLE
q
r
p
Given: The wheel consists of a thin ring
with a mass 10 kg and four spokes
(slender rods) with a mass 2 kg
each.
Find: The wheel’s MMI about an axis
perpendicular to the screen and
passing through point A.
Plan:
Follow steps similar to finding the
MoI for a composite area.
Solution:
1. The wheel can be divided into a thin ring (p) and two slender
rods (q and r). Will both rods be treated the same?
EXAMPLE (continued)
2. The center of mass for each of the three
pieces is at point O, 0.5 m from Point A.
q
r
O
p
3. The MMI data for a thin ring and
slender rod are given on the inside
back cover of the textbook. Using
those data and the parallel-axis
theorem, calculate the following.
IA = IO + (m) (d) 2
IAp = 10 (0.5)2 + 10 (0.5)2 = 5.0 kg·m2
IAq = IAr = (1/12) (4) (1)2 + 4 (0.5)2 = 1.333 kg·m2
4. Now add the three MMIs about point A.
IA = IAp + IAq + IAr = 7.67 kg·m2
CONCEPT QUIZ
1. Consider a particle of mass 1 kg
z
located at point P, whose coordinates
·P(3,4,6)
are given in meters. Determine the MMI
of that particle about the z axis.
y
A) 9 kg·m2
B) 16 kg·m2
x
C) 25 kg·m2
D) 36 kg·m2
2. Consider a rectangular frame made of four
slender bars with four axes (zP, zQ, zR and zS)
P
Q
perpendicular to the screen and passing
•
•
through the points P, Q, R, and S respectively.
S•
•R
About which of the four axes will the MMI
of the frame be the largest?
A) zP
B) zQ
C) zR
D) zS
E) Not possible to determine.
GROUP PROBLEM SOLVING
R
P
Plan:
Given: The pendulum consists of a 24 lb
plate and a slender rod weighing
8 lb.
Find: The radius of gyration of
the pendulum about an axis
perpendicular to the screen and
passing through point O.
Determine the MMI of the pendulum using the method for
composite bodies. Then determine the radius of gyration
using the MMI and mass values (check units!!).
Solution
1. Separate the pendulum into a square plate (P) and a slender
rod (R).
GROUP PROBLEM SOLVING
R
P
2. The center of mass of the plate
and rod are 3.5 ft and 0.5 ft from
point O, respectively.
3. The MMI data on plates and slender rods are given on the
inside cover of the textbook. Using those data and the parallel-axis
theorem,
IP = (1/12) (24/32.2) (12 + 12) + (24/32.2) (3.5)2 = 9.254 slug·ft2
IR = (1/12) (8/32.2) (5)2 + (8/32.2) (0.5)2 = 0.5797 slug·ft2
4. IO = IP + IR = 9.254 + 0.5797 = 9.834 slug·ft2
5. Total mass (m) equals (24+8)/32.2 = 0.9938 slug
Radius of gyration k = IO / m = 3.15 ft
ATTENTION QUIZ
1. A particle of mass 2 kg is located 1 m
down the y-axis. What are the MMI of
the particle about the x, y, and z axes,
respectively?
x
A) (2, 0, 2)
B) (0, 2, 2)
C) (0, 2, 2)
D) (2, 2, 0)
2. Consider a rectangular frame made of four
slender bars and four axes (zP, zQ, zR and zS)
perpendicular to the screen and passing
through points P, Q, R, and S, respectively.
About which of the four axes will the
MMI of the frame be the lowest?
A) zP
B) zQ
C) zR
D) zS
E) Not possible to determine.
z
1m
•
P
•
S•
y
Q
•
•R