A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

Video transcript

Hello everyone in this problem, we have a metal piece of mass M, which is extended to form a plainer rectangular sheet. The length of the sheet, L is twice the width. Find the moment of inertia about an axis perpendicular to the sheet and passing through one of the four corners. So we have a metal sheet. Mass is equal to um the length is L. Or the width. It's hell over two. So we know at the moment of inertia passing through the central access Lehner sheet, I see em centre of mass is equal to 1 and times the length squared pasta with square. This question is asking us to find the moment of inertia about access perpendicular to the sheet passing through one of the four corners of the axis of rotation. Is rather here instead of the center of mass. So instead we could use the parallel axis theorem which says that I prime is equal to I the center of mass plus and times the distance from the center of mass two, the axis of rotation. So this is what we are R C. M. To define this, we can use the pythagorean theorem. We know that this whole section is out, So this must be L over two. We know this, this whole section. It's all over too. So this height must be out before so we have this triangle here when this hippopotamuses, R C. M. This side is out over to this side is out over four. We could use pythagorean theorem to calculate for R C. M equal to the square root. Well over four squared plus over two squared, solve this and simplify. We get The RCM is five squared over 16, so we have our CM. And we need to simplify our equation for I C M. Call at L squared was W where W is L over two Times The Mass Times 1. 12 gives us I see, em. And we can add this M times the square root. Five L squared over 16 square term. I mean, do this simple simplification. We get that I prime is equal to 5/12 M elsewhere with his answer choice. See hope this helps have a great day.