MOMENT OF INERTIA

1. MOMENT OF INERTIA BY GP CAPT NC CHATTOPADHYAY

2. WHAT IS MOMENT OF INERTIA? • IT IS THE MOMENT REQUIRED BY A SOLID BODY TO OVERCOME IT’S RESISTANCE TO ROTATION • IT IS RESISTANCE OF BENDING MOMENT OF A BEAM • IT IS THE SECOND MOMENT OF MASS (mr2) OR SECOND MOMENT OF AREA (Ar2) • IT’S UNIT IS m4 OR kgm2

3. PERPENDICULAR AXIS THEOREM • The moment of inertia of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis.  Moment of Inertia: Iz = Ix+Iy

4. PARALLEL (TRANSFER)AXIS THEOREM • THE MOMENT OF AREA OF ANOBJECT ABOUT ANY AXIS PARALLEL TO THE CENTROIDAL AXIS IS THE SUM OF MI ABOUT IT’S CENTRODAL AXIS AND THE PRODUCT OF AREA WITH THE SQUARE OF DISTANCE OF CG FROM THE REF AXIS • IXX= IG+Ad2 • A is the cross-sectional area. : is the perpendicuar distance between the centroidal axis and the parallel axis.

5. Moment of Inertia - Parallel Axis Theorem Parallel axis theorem: Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y the distance from an element of area dA to AA’.

6. Moment of Inertia - Parallel Axis Theorem Consider an axis BB’ parallel to AA’ through the centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes

7. Moment of Inertia - Parallel Axis Theorem The first integral is the moment of inertia about the centroid. The second component is the first moment area about the centroid

8. Moment of Inertia - Parallel Axis Theorem Modify the equation obtained with the parallel axis theorem.

9. Example – Moment of Inertia Compute the moment of inertia in the x about the AA` plane. AA`

10. AA` Example – Moment of Inertia Compute the moment of inertia in the x about the AA` plane.

11. Example – Moment of Inertia From earlier lecture, the moment of inertia about the centroid

12. AA` Example – Moment of Inertia Using the parallel axis theorem

13. Parallel Axis - Why? Recall that the method of finding centroids of composite bodies? - Follow a Table technique How would you be able to find the moment of inertia of the body. Use a similar technique, table method, to find the moment of inertia of the body.

14. Parallel Axis - Why? Use a similar technique, table method, to find the moment of inertia of the body.

15. Moment of Inertia Use a set of standard tables:

16. Example - Moment of Inertia Find the moment of inertia of the body, Ix and the radius of gyration, kx (rx)

17. I THINK…. I CAN THINK….

18. Example - Moment of Inertia Set up the reference axis at AB and find the centroid

19. Example - Moment of Inertia From the table to find the moment of inertia

20. Example - Moment of Inertia Compute the radius of gyration, rx.

21. Example - Moment of Inertia Find the moment of inertia of the body, Ix and the radius of gyration, kx (rx)

22. Example - Moment of Inertia The components of the two bodies and subtract the center area from the total area.

23. Example - Moment of Inertia Compute the radius of gyration, rx.

24. PROBLEM…. • FIND THE MI OF A CIRCULAR LAMINA OF RADIUS “r“?