Mass & Inertia Session 03-06
-
Upload
mallory-daniels -
Category
Documents
-
view
27 -
download
0
description
Transcript of Mass & Inertia Session 03-06
![Page 1: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/1.jpg)
Mass & InertiaSession 03-06
Subject : S1014 / MECHANICS of MATERIALSYear : 2008
![Page 3: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/3.jpg)
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.
![Page 4: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/4.jpg)
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.
![Page 5: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/5.jpg)
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 = mgSince the weight is a force, its SI unit is the newton. Density is mass/volume. …..
![Page 6: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/6.jpg)
Bina Nusantara
What is Inertia ?
Inertia is the
resistance an object
has to a change in its state of motion.
![Page 7: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/7.jpg)
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
![Page 8: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/8.jpg)
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.
![Page 9: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/9.jpg)
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 = mr2
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.
![Page 10: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/10.jpg)
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.
![Page 11: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/11.jpg)
Bina Nusantara
What is Moment Inertia ?
![Page 12: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/12.jpg)
Bina Nusantara
Moment Inertia
http://hyperphysics.phy-astr.gsu.edu/hbase/inecon.html
![Page 13: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/13.jpg)
Bina Nusantara
SOLID CYLINDER
![Page 14: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/14.jpg)
Bina Nusantara
SOLID CYLINDER
Central Axismoment of inertia:
The mass element can be expressed in terms
Substituting gives a polynomial form integral:
![Page 15: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/15.jpg)
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.
![Page 16: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/16.jpg)
Bina Nusantara
Hoop AboutSymmetry Axis
![Page 17: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/17.jpg)
Bina Nusantara
Hoop AboutSymmetry Axis
![Page 18: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/18.jpg)
Bina Nusantara
SOLID SPHERESymmetry Axis
![Page 19: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/19.jpg)
Bina Nusantara
SOLID SPHERESymmetry Axis
![Page 20: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/20.jpg)
Bina Nusantara
ROD ABOUTCenter
![Page 21: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/21.jpg)
Bina Nusantara
ROD ABOUTCenter
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
![Page 22: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/22.jpg)
Bina Nusantara
Moment of Inertia
axis through center
axis through end
![Page 23: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/23.jpg)
Bina Nusantara
Moment of Inertia
axis through center
axis through end
![Page 24: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/24.jpg)
Bina Nusantara
Moment of Inertia
thin-walled hollow
solid
![Page 25: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/25.jpg)
Bina Nusantara
Moment of Inertia
hollow
solid
![Page 26: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/26.jpg)
Bina Nusantara
Moment of Inertia
thin-walled hollow
![Page 27: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/27.jpg)
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.
![Page 28: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/28.jpg)
Bina Nusantara
Radius of Gyration
It is possible for two EQUAL areas to have equal MOMENT'S OF INERTIA about the same axis while having different dimensions.
h1 and b1 are constants.
b2 depends on h2
d is maximized when h2 goes to zero
![Page 29: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/29.jpg)
Bina Nusantara
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.
![Page 30: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/30.jpg)
Bina Nusantara
Area Radius of Gyration
The Radius of Gyration kx of an Area (A) about an axis (x) is defined as
Where Ix is the Moment of Inertia about the axis (x), and A is the area
kx
![Page 31: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/31.jpg)
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
![Page 32: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/32.jpg)
Bina Nusantara
Mass Radius of Gyration
The Radius of Gyration kxx of a Mass (m) about an axis (x) is defined as:
Where I is the Moment of Inertia about the axis (x), and m is the mass.
kxx
![Page 33: Mass & Inertia Session 03-06](https://reader036.fdocuments.net/reader036/viewer/2022062422/56812d5f550346895d926bd5/html5/thumbnails/33.jpg)
Bina Nusantara
Inertia & Radius of Gyration
http://www.efunda.com/math/areas/rectangle.cfm