Lec5 Rigid Dynamics

1. Rigid Body Dynamics
GSE 264
Assoc. Prof. Mohamed Shiba

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

3. 17.1 Mass Moment of Inertia
We define the moment of inertia as the integral of the “second moment” about an axis of all
the elements of mass dm which compose the body

4. 17.1 Mass Moment of Inertia

6. 17.1 Mass Moment of Inertia
Example:

7. 17.1 Mass Moment of Inertia

8. 17.1 Mass Moment of Inertia
Parallel-Axis Theorem
If the moment of inertia of the body about an axis passing through the
body’s mass center is known, then the moment of inertia about any other
parallel axis can be determined by using the parallel-axis theorem.

9. 17.1 Mass Moment of Inertia
Radius of Gyration:
Occasionally, the moment of inertia of a body about a specified axis is reported in
handbooks using the radius of gyration, k.
If the body mass is condensed in one point, the required perpendicular distance
between a given axis and the mass point to have the same mass moment of inertia as
that of the body is called radius of gyration
Composite Bodies:
If a body consists of a number of simple shapes such as disks, spheres, and rods, the
moment of inertia of the body about any axis can be determined by adding algebraically
the moments of inertia of all the composite shapes computed about the axis.

11. Example:
If the plate shown in Fig. has a density of 8000 kg/m3 and a thickness of 10 mm,
determine its moment of inertia about an axis directed perpendicular to the page and
passing through point O.
17.1 Mass Moment of Inertia

12. Example:
If the plate shown in Fig. has a density of 8000 kg/m3 and a thickness of 10 mm,
determine its moment of inertia about an axis directed perpendicular to the page and
passing through point O.
17.1 Mass Moment of Inertia

13. Example:
The pendulum in Fig. 17–7 is suspended from the pin at O and
consists of two thin rods. Rod OA weighs 10 lb, and BC weighs 8
lb. Determine the moment of inertia of the pendulum about an axis
passing through (a) point O, and (b) the mass center G of the
pendulum.
17.1 Mass Moment of Inertia

14. Example:
The pendulum in Fig. 17–7 is suspended from the pin at O and
consists of two thin rods. Rod OA weighs 10 lb, and BC weighs 8
lb. Determine the moment of inertia of the pendulum about an axis
passing through (a) point O, and (b) the mass center G of the
pendulum.
17.1 Mass Moment of Inertia
SOLUTION
Part (a). Using the table, the moment of inertia of rod OA about an
axis perpendicular to the page and passing through point O of the
rod is

15. Example:
The pendulum in Fig. 17–7 is suspended from the pin at O and
consists of two thin rods. Rod OA weighs 10 lb, and BC weighs 8
lb. Determine the moment of inertia of the pendulum about an axis
passing through (a) point O, and (b) the mass center G of the
pendulum.
17.1 Mass Moment of Inertia
SOLUTION

16. Example:
The pendulum consists of a disk having
a mass of 6 kg and slender rods AB and
DC which have a mass per unit length of
2 kg/m. Determine the length L of DC so
that the center of mass is at the bearing
O. What is the moment of inertia of the
assembly about an axis perpendicular to
the page and passing through point O?
17.1 Mass Moment of Inertia

17. Example:
The pendulum consists of a disk having
a mass of 6 kg and slender rods AB and
DC which have a mass per unit length of
2 kg/m. Determine the length L of DC so
that the center of mass is at the bearing
O. What is the moment of inertia of the
assembly about an axis perpendicular to
the page and passing through point O?
17.1 Mass Moment of Inertia
SOLUTION

18. Assignments
17–11. The assembly is made of the slender rods that have a mass per unit length
of 3 kg/m. Determine the mass moment of inertia of the assembly about an axis
perpendicular to the page and passing through point O.
17.1 Mass Moment of Inertia

19. Assignments
17–13. The wheel consists of a thin ring having a mass of 10 kg and four spokes
made from slender rods and each having a mass of 2 kg. Determine the wheel’s
moment of inertia about an axis perpendicular to the page and passing through point
A.
17.1 Mass Moment of Inertia

20. Assignments
17–18. Determine the moment of inertia of the assembly about an axis
perpendicular to the page and passing through point O. The block has a mass of 3
kg, and the semicylinder has a mass of 5 kg.
17.1 Mass Moment of Inertia

21. Assignments
17.1 Mass Moment of Inertia

22. Faculty of
Engineering