Problem?(a) For the thin rectangular plate shown in part (d) of Table 9.2, find the moment of inertia about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis shown. (b) Find the moment of inertia of the plate for an axis that lies in the plane of the plate, passes through the center of the plate, and is perpendicular to the axis in part (a).

Solution 56E Problem (a) Step 1: To find the moment of inertia about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis. Consider axis shown in blue colour be the origin Mass of the rectangular plate = M Breadth of the plate = a a The axis of rotation lies at the distance 2 from its origin and center of mass passes through the axis of rotation. Step 2: We can apply parallel axis theorem to solve this problem I = ICM+ MR 2 a Here R = 2 I = Ma 2 3 Step 3: Therefore the moment of inertia about the axis through which the center of mass passes I = Ma - Ma 1 2 CM 3 4 Simplifying the above equation 1 2 ICM= 12Ma Problem (b) Step 1: To Find the moment of inertia of the plate for an axis that lies in the plane of the plate, passes through the center of the plate, and is perpendicular to the axis. The problem is same as that of the previous problem. There is a small change here. Breadth a is changed as length b. Mass of the plate = M Length of the plate = b b The axis of rotation lies at the distance 2from its origin and center of mass passes through the axis of rotation. Step 2: Applying parallel axis theorem I = ICM+ MR 2 b Here R = 2 1 2 I = M3