1. Chapter 16. Kinetics of Rigid Bodies: Forces And Accelerations
Engr 240 – Week 12
Chapter 16. Kinetics of Rigid Bodies: Forces And Accelerations

2. Equations of Motion for a Rigid Body
and .
The system of the external forces is equivalent to the system consisting of the vector attached at G and the couple of moment

3. Angular Momentum In Plane Motion
Consider a rigid slab made up of a large number of particles of mass
The angular momentum about the centroidal frame Gx’y’ is
where:
- position vector of ith particle
- linear momentum of ith particle
But:
and of magnitude
Therefore
which has the same direction as
Hence:
Centroidal moment of inertia to the slab.
Differentiating both sides with respect to time:

4. D’ALEMBERT’S PRINCIPLE
For a rigid body in plane motion:
and ,
The external forces acting on a rigid body is equivalent to the system consisting of a vector attached to the center of mass G, and a couple of moment .
The two vector equations: three independent scalar equations:
and

5. Translation:
Centroidal Rotation:
NOTE: Before applying D’Alembert’s principle, it may be necessary to analyze kinematics of motion to reduce the number of unknown kinematic variables.

6. Example 1.
A pulley weighing 12 lb and having a radius of gyration of 8 in is connected to two blocks as shown. Assuming no axle friction, determine the angular acceleration of the pulley and the acceleration of each block.

7. Example 2. The spool has a mass of 8 kg and a radius of gyration of kG=0.35 m about its center. If cords of negligible mass are wrapped around its inner hub and outer rim as shown, determine its angular acceleration.

8. Example 3. A slender bar AB weighs 60 lbs and moves in the vertical plane, with its ends constrained to follow the smooth horizontal and vertical guides. If the 30-lb force is applied at A with the bar initially at rest in the position shown for which =30, calculate the resulting angular acceleration of the bar and the forces on the small end rollers at A and B.

10. Rolling Motion:
Rolling Without Sliding

11. Rolling With Sliding: