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MASS MOMENT OF INERTIA

Todays Objectives Students will be able to a)

Explain the concept of the mass moment of

inertia (MMI). b) Determine the MMI of a

composite body.

- In-Class Activities
- 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

What property of the flywheel is most important

for this use? How can we determine a value for

this property?

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.

Why is most of the mass of the flywheel located

near the flywheels 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 (Section 10.9)

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

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.

p

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 through the bodys

center of mass G. The formula is Ip IG

(m) (d)2 (where m is the mass of the body).

d

p

G

m

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.

EXAMPLE

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 wheels 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.

q

r

p

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.

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.

O

IA IO (m) (d) 2 IAp 10

(0.5)2 10 (0.5)2 5.0 kgm2 IAq

IAr (1/12) (4) (1)2 4 (0.5)2 1.333

kgm2 4. Now add the three MMIs about point A.

IA IAp IAq IAr 7.67 kgm2

CONCEPT QUIZ

1. Consider a particle of mass 1 kg located

at point P, whose coordinates are given in

meters. Determine the MMI of that particle

about the z axis. A) 9 kgm2

B) 16 kgm2 C) 25 kgm2

D) 36 kgm2

2. Consider a rectangular frame made of four

slender bars with four axes (zP, zQ, zR and

zS) perpendicular to the screen and passing

through the points P, Q, R, and S respectively.

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

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.

P

Plan 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

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 slugft2 IR (1/12)

(8/32.2) (5)2 (8/32.2) (0.5)2 0.5797 slugft2

4. IO IP IR 9.254 0.5797 9.834

slugft2

5. Total mass (m) equals (248)/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? 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.

End of the Lecture

Let Learning Continue

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