Discussion and hints:
If the shaft supporting the disk passed through the center of mass C of the disk, then the reaction force at the shaft would be simply equal and opposite of the weight of the disk. With the shaft attached at point O, a “dynamic imbalance” is created at the bearings of the shaft. The rotation of the shaft will produce dynamic loads at the bearings. Your task here is to calculate those support forces resulting from the dynamic imbalance.
Recall the following four-step plan outline in the lecture book and discussed in lecture:
Step 1: FBDs
Draw a free body diagram of the disk.
Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the disk using your FBD above. Take care in choosing the reference point for your moment equation. In order to use the “short form” of Euler’s equation, this point should be either a fixed point or the body’s center of mass. For this problem, you are able to choose either O or C for your reference point.
Step 3: Kinematics
Note that the center of mass of the disk C travels on a circular path of radius r/2 with a constant speed. You can use either a polar description, a path description or the rigid body equations for the kinematics of of C in your analysis.
Step 4: Solve
Solve your equations above for the x– and y-components of the reactions on the disk at the shaft O: these forces are indicated as Ox and Oy in the above animation of the motion.
Any questions?