A thin, uniform rod is bent into a square of side length ?a?. If the total mass is ?M?, find the moment of inertia about an axis through the center and perpendicular to the plane of the square. ? int:? Use the parallel-axis theorem.)

Solution 96P Introduction First we will the moment of inertia of the each side of the square about an axis passing through the center of the square. Now using parallel axis theorem we will calculate the moment of inertia of the side about the center of the square. Now the moment inertia of the square will be the four times the moment of inertia of one side. Step 1 If we make square from a wire of length a, then the length of the each side of the square will be L = 4 And the mass of the each side of the square will be M m = 4 Now we can consider each side of the square as a rod. Now we will calculate the moment of inertia of the rod about the and axis perpendicular to the plane of the square and passing through the center of the rod. Hence the moment of inertia is 2 Icm = 1 mL = 1 ( )( )a = 1 Ma 2 12 12 4 4 768 Step 2 Now the distance from the center of the center of the side is the half of the one side fo the square, hence we can write that d = a 8 Now the moment of inertia of the side about the center of the square is I = I + md = 1 Ma +2 M a 2 = 1 Ma + 1 Ma =2 1 Ma 2 c cm 768 4 (8) 768 256 192