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King Fahd University of Petroleum & Minerals

This lecture discusses the moment of inertia of a body, parallel axis theorem, radius of gyration, moment and angular acceleration, and moment of inertia of composite bodies.

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King Fahd University of Petroleum & Minerals

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  1. King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 28

  2. CHAPTER 17 Planar Kinetics of a Rigid Body: Force and Acceleration

  3. Objective • Moment of Inertia of a body • Parallel Axis Theorem • Radius of Gyration • Moment of Inertia of Composite Bodies

  4. Moment and Angular Acceleration • When M  0, rigid body experiences angular acceleration • Relation between M or (t) and a is analogous to relation between F and a Moment of Inertia Mass = Resistance

  5. Moment of Inertia • This mass analog is called the moment of inertia, I, of the object • r = moment arm • SI units are kg m2

  6. Moment of Inertia 4m 3m 2a m 2m a • Calculate I about horizontal axis through center • I depends on both the mass and its distribution. • If an object’s mass is distributed further from the axis of rotation, the moment of inertia will be larger.

  7. Moment of Inertia 4m 3m 2a m 2m a • Calculate I about vertical axis through center

  8. Shell Element Disk Element

  9. Example 17-1

  10. Example 17-2

  11. L R2 R2 R a R b R R Moments of inertia for some common geometric solids L a b

  12. Cylindrical shell about axis R Moments of inertia Hollow cylinder about axis R1 R2 L Solid cylinder about axis Cylindrical shell about diameter through center R R L

  13. Solid cylinder about diameter through center R L Moments of inertia Thin rod about perpendicular line through center L Thin spherical shell about diameter R Solid sphere about diameter R

  14. Parallel Axis Theorem • The moment of inertia about any axis parallel to and at distance d away from the axis that passes through the centre of mass is: • Where • IG= moment of inertia for mass centre G • m = mass of the body • d = perpendicular distance between the parallel axes.

  15. Radius of Gyration Frequently tabulated data related to moments of inertia of areas will be presented in terms of radius of gyration. Since this parameter is defined to by the following formula it may be used to compute the moment of inertia of the area provided the area is known or may be computed.

  16. Mass Center Example

  17. Moment of Inertia of Composite bodies • Divide the composite area into simple body. • Compute the moment of inertia of each simple body about its centroidal axis from table. • Transfer each centroidal moment of inertia to a parallel reference axis by the use of formula. • The sum of the moments of inertia for each simple body about the parallel reference axis is the moment of inertia of the composite body. Any cutout area has must be assigned a negative moment; all others are considered positive.

  18. Moment of inertia of a hollow cylinder • Moment of Inertia of a solid cylinder • A hollow cylinder I = 1/2 mR2 - = m2 M m1 R1 R2 I = 1/2 m1R12 - 1/2 m2R22 = 1/2 M (R12 + R22 )

  19. Example 17-3

  20. Example 17-4

  21. Thank You

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