Subject
Year
: S1014 / MECHANICS of MATERIALS
: 2008
Mass & Inertia
Session 03-06
Massa & Inertia
Bina Nusantara
What is Mass ?
Mass is such a fundamental quantity
that it is hard to define in terms of something
else.
All mechanical quantities can be defined
in terms of mass, length, and time.
Bina Nusantara
What is Mass ?
The usual symbol for mass is
m and
its SI unit is the kilogram.
While the mass is normally considered to be an
unchanging property of an object, at speeds
approaching the speed of light one must consider
the increase in the relativistic mass.
Bina Nusantara
What is the deference between Mass vs Weight ?
…The weight of an object is the force of gravity
on the object and may be defined as the mass
times the acceleration of gravity,
w = mg
Since the weight is a force, its SI unit is the
newton. Density is mass/volume. …..
Bina Nusantara
What is Inertia ?
Inertia is the
resistance an
object has to a change in its
state of motion.
Bina Nusantara
What is Inertia ?
The principle of inertia is one of the fundamental principles of
classical physics which are used to
describe the
motion of matter and how it is affected by applied forces
Bina Nusantara
What is Moment Inertia ?
Moment of inertia is the name given to
rotational inertia, the rotational analog of
mass for linear motion.
It appears in the relationships for the
dynamics of rotational motion.
The moment of inertia must be specified with
respect to a chosen axis of rotation.
Bina Nusantara
What is Moment Inertia ?
For a point mass the moment of inertia is just the
mass times the square of perpendicular distance
to the rotation axis,
I=
2
mr
That point mass relationship becomes the basis
for all other moments of inertia since any object
can be built up from a collection of point masses.
Bina Nusantara
What is Moment Inertia ?
Moment of inertia is defined with respect to a specific
rotation axis.
The moment of inertia of a point mass with respect to an axis is
defined as the product of the mass times the distance from the
axis squared.
The moment of inertia of any extended object is built up from that
basic definition. The general form of the moment of inertia
involves an integral.
Bina Nusantara
What is Moment Inertia ?
Bina Nusantara
Moment Inertia
Bina Nusantara
http://hyperphysics.phy-astr.gsu.edu/hbase/inecon.html
SOLID CYLINDER
Bina Nusantara
SOLID CYLINDER
Central Axis
moment of inertia:
The mass element can be expressed in
terms
Substituting gives a polynomial form
integral:
Bina Nusantara
SOLID CYLINDER
Central Diameter
Now expressing the mass element dm
in terms of z, we can integrate over
the length of the cylinder.
Bina Nusantara
Hoop About
Symmetry Axis
Bina Nusantara
Hoop About
Symmetry Axis
Bina Nusantara
SOLID SPHERE
Symmetry Axis
Bina Nusantara
SOLID SPHERE
Symmetry Axis
Bina Nusantara
ROD ABOUT
Center
Bina Nusantara
ROD ABOUT
Center
Since the total length L has mass
M, then M/L is the proportion of
mass to length and the mass
element can be expressed as
shown. Integrating from -L/2 to
+L/2 from the center includes the
entire rod
Bina Nusantara
Moment of Inertia
axis through center
axis through end
Bina Nusantara
Moment of Inertia
axis through center
axis through end
Bina Nusantara
Moment of Inertia
thin-walled hollow
solid
Bina Nusantara
Moment of Inertia
hollow
solid
Bina Nusantara
Moment of Inertia
thin-walled hollow
Bina Nusantara
What is Gyration ?
Radius of gyration is the name of
several related measures of the
size of an object, a surface, or an
ensemble of points.
Bina Nusantara
Radius of Gyration
h1 and b1 are constants.
b2 depends on h2
It is possible for two EQUAL areas to have
equal MOMENT'S OF INERTIA about the
same axis while having different
dimensions.
Bina Nusantara
d is maximized when h2 goes to zero
Area Radius of Gyration
• The Radius of Gyration of an Area about a
given axis is a distance k from the axis.
• At this distance k an equivalent area is
thought of as a line Area parallel to the
original axis.
• The moment of inertia of this Line Area about
the original axis is unchanged.
Bina Nusantara
Area Radius of Gyration
The Radius of Gyration kx of an Area (A) about an axis (x) is defined
as
kx
Where Ix is the Moment of Inertia about the axis (x), and A is the area
Bina Nusantara
Mass Radius of Gyration
•The Radius of Gyration of a Mass about a
given axis is a distance k from the axis.
•At this distance k an equivalent mass is
thought of as a Point Mass.
•The moment of inertia of this Point Mass
about the original axis is unchanged
Bina Nusantara
Mass Radius of Gyration
The Radius of Gyration kxx of a Mass (m) about an axis (x) is defined
as:
kxx
Where I is the Moment of Inertia about the axis (x), and m is the mass.
Bina Nusantara
Inertia & Radius of Gyration
http://www.efunda.com/math/areas/rectangle.cfm
Bina Nusantara