1. Center of Gravity The balance point of an object.
Section 5
Center of Gravity
The balance point of an object.
For many simple parts, such as a cylinder or block, the geometric center is apparent.

2. Complex Parts The center of gravity of complex parts can be
determined form a weighted
average of the coordinates of the individual cg’s.

3. Problem 5-1 The plate shown is made from steel
(0.283 lb/in3). Determine the coordinates of the center of gravity. 6” 3”
12”
2”
0.5” x z y

4. Mass Moment of Inertia, I
Resistance to rotational acceleration.
Computed relative to an axis
Strongly influenced by the amount of mass distributed from the axis. r dm A

5. Mass Moment of Inertia Charts available for common shapes. Cylinder: yx z y

6. Radius of Gyration, k
Distance from the reference axis, to a point, where a concentrated mass would have the same moment of inertia.
Occasionally used in dynamic testing.

7. Problem 5-3
Calculate the mass moment of inertia and the radius of gyration about a centroidal longitudinal axis of a shaft that weighs 5 lb and has a diameter of in. d

8. Problem 5-5:
A solid cylinder is 2 ft in diameter, 3 ft long and weighs 48 lbs. Determine the mass moment of inertia about its centroidal axial axis.

9. Parallel Axis Theorem
Can transfer the moment of inertia, from one axis to another.
IA’ = IA  m(d)2
d is the distance between the two axes
Add if transfer is away from the centroid.
Subtract if transfer is towards the centroid.

10. Problem 5-7:
A slender rod, 14 in. long, rotates about an axis perpendicular to its length and 3 inches from its center of gravity. Knowing that the rod weighs 2 lb, determine its mass moment of inertia about that axis. d

11. Composite Body
The moment of inertia of a body, comprised of several simple shapes, can be combined as long as there is a single reference axis.
IA (Total) = IA (Body 1) + IA (Body 2) +…

12. Problem 5-14
Determine the mass moment of inertia of the plate about the rotation axis. The plate is made from steel, with a density of
0.281 lb/in3.
3.5 in
4.0 in
2 in
0.5 in
0.75 in