The document is a set of lecture notes about centroid, center of gravity, and moment of inertia. It discusses how the centroid and center of gravity of uniform objects like rods can be found, and how they are defined in terms of the first moment of the object. It also covers finding the center of gravity and center of mass for systems of particles and composite bodies, as well as definitions and calculations related to moments of inertia for areas, including moments of inertia for inclined axes.

1. “Lecture notes on”
UNIT- 1 (A)
CENTROID ,CENTRE OF GRA
VITYAND MOMENT OF INERTIA
MECHANICS OF MATERIALS
By
Sougata Das
Lecturer in CE/JGP

2. CENTER OF MASS AND CENTER OF GRAVITY
• Consider a uniform rod of length L. When the rod is placed horizontally, each
particle of the rod is pulled by the gravitational force of the earth. The sum of all
three forces will be equal to the total weight of the rod.
• Consider two systems: One is the rod AB of weight W and other is a concentrated
weight W, at a distance 1/3 Lfrom point A.
•These two systems are equivalent as far as the net gravitational pull is concerned, but
the moment of the gravitational forces of the two systems about point Ais different.

3. CENTER OF MASS AND CENTER OF GRAVITY
• If a concentrated weight of W is put at a distance of 1/2 L from point A, then the
moment of gravitational pull in uniform rod as well as the new system will be same.
•In this case, not only the weight but the first moment of an element of a quantity
about an origin is defined d as the product of its position vector with the element
physical quantity itself.
•With the concept of first moment, we define the centroid centre of mass and centre of
gravity.

4. CENTER OF MASS AND CENTER OF GRAVITY

5. CENTER OF MASS AND CENTER OF GRAVITY

6. CENTER OF MASS AND CENTER OF GRAVITY

7. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

8. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

9. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

10. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

11. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

12. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

13. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

14. CENTER OF GRAVITYAND CENTER OF MASS FOR ASYSTEM OF PARTICLES

15. EXAMPLE

16. EXAMPLE

17. EXAMPLE

18. COMPOSITE BODIES

19. COMPOSITE BODIES

20. COMPOSITE BODIES

21. COMPOSITE BODIES

22. SOLUTION

23. SOLUTION

24. DEFINITION OF MOMENTS OF INERTIA FOR AREAS

25. DEFINITION OF MOMENTS OF INERTIA FOR AREAS

26. DEFINITION OF MOMENTS OF INERTIA FOR AREAS

27. PARALLELAXIS THEOREM FOR AN AREA

28. PARALLELAXIS THEOREM FOR AN AREA

29. PARALLELAXIS THEOREM FOR AN AREA

30. RADIUS OF GYRATION OF AN AREA

31. EXAMPLE

32. SOLUTION

33. SOLUTION

34. MOMENTS OF INERTIA FOR COMPOSITE AREAS

35. MOMENTS OF INERTIA FOR COMPOSITE AREAS

36. EXAMPLE

37. SOLUTION

38. SOLUTION

39. SOLUTION

40. PRODUCT OF INERTIA FOR AN AREA

41. PRODUCT OF INERTIA FOR AN AREA

42. EXAMPLE

43. SOLUTION

44. SOLUTION

45. SOLUTION

46. MOMENTS OF INERTIA FOR AN AREA ABOUT
INCLINED AXES

47. MOMENTS OF INERTIA FOR AN AREA ABOUT
INCLINED AXES

48. MOMENTS OF INERTIA FOR AN AREA ABOUT
INCLINED AXES

49. MOMENTS OF INERTIA FOR AN AREA ABOUT
INCLINED AXES

50. MOMENTS OF INERTIA FOR AN AREA ABOUT
INCLINED AXES