1. Mass & Inertia Session 03-06
Subject : S1014 / MECHANICS of MATERIALS
Year : 2008
Mass & Inertia Session 03-06

2. Massa & Inertia
Bina Nusantara

3. 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

4. The usual symbol for mass is m and its SI unit is the kilogram.
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

5. 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

6. What is Inertia ?
Inertia is the resistance an object has to a change in its state of motion.
Bina Nusantara

7. 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

8. It appears in the relationships for the dynamics of rotational motion.
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

9. What is Moment Inertia ? I = mr2
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.
Bina Nusantara

10. 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

11. What is Moment Inertia ?
Bina Nusantara

12. Moment Inertia http://hyperphysics.phy-astr.gsu.edu/hbase/inecon.html
Bina Nusantara

13. SOLID CYLINDER
Bina Nusantara

14. SOLID CYLINDER Central Axis moment of inertia:
                                   
The mass element can be expressed in terms                                                             
Substituting gives a polynomial form integral:
                                                                           
Bina Nusantara

15. 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

16. Hoop About
Symmetry Axis
Bina Nusantara

17. Hoop About
Symmetry Axis
Bina Nusantara

18. SOLID SPHERE Symmetry Axis
Bina Nusantara

19. SOLID SPHERE Symmetry Axis
Bina Nusantara

20. ROD ABOUT Center
Bina Nusantara

21. 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

22. Moment of Inertia
axis through center
axis through end
Bina Nusantara

23. Moment of Inertia
axis through center
axis through end
Bina Nusantara

24. Moment of Inertia
thin-walled hollow
solid
Bina Nusantara

25. Moment of Inertia
hollow
solid
Bina Nusantara

26. Moment of Inertia
thin-walled hollow
Bina Nusantara

27. 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

28. 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.
d is maximized when h2 goes to zero
Bina Nusantara

29. 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

30. 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

31. 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

32. 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

33. Inertia & Radius of Gyration
Bina Nusantara