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

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Presentation on theme: "Mass & Inertia Session 03-06 Subject: S1014 / MECHANICS of MATERIALS Year: 2008."— Presentation transcript:

1 Mass & Inertia Session Subject: S1014 / MECHANICS of MATERIALS Year: 2008

2 Bina Nusantara Massa & Inertia

3 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.

4 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.

5 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. …..

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

7 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

8 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.

9 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 = mr 2 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.

10 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.

11 Bina Nusantara What is Moment Inertia ?

12 Bina Nusantara Moment Inertia

13 Bina Nusantara SOLID CYLINDER

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

15 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.

16 Bina Nusantara Hoop About Symmetry Axis

17 Bina Nusantara Hoop About Symmetry Axis

18 Bina Nusantara SOLID SPHERE Symmetry Axis

19 Bina Nusantara SOLID SPHERE Symmetry Axis

20 Bina Nusantara ROD ABOUT Center

21 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

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

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

24 Bina Nusantara Moment of Inertia thin-walled hollow solid

25 Bina Nusantara Moment of Inertia hollow solid

26 Bina Nusantara Moment of Inertia thin-walled hollow

27 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.

28 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

29 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.

30 Bina Nusantara Area Radius of Gyration The Radius of Gyration k x 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 kxkx

31 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

32 Bina Nusantara Mass Radius of Gyration The Radius of Gyration k xx 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. k xx

33 Bina Nusantara Inertia & Radius of Gyration


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