A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door’s moment of inertia for (b) rotation about a vertical axis inside the door, 15 cm from one edge?

Video transcript

Welcome back everyone in this problem, a wooden plank with a mass of 15 kg as dimensions as follows a length of 180 centimeters and width of 75 centimeters. If it rotates around an axis that is parallel to its length, located at a distance of just 10 centimeters away from one edge and is inside the plank. What would be its moment of inertia? A says it's 1.8 kg square meters. B 3.9 kg square meters c 6.3 kg square meters and the d 9.6 kg square meters. Now, if we are going to figure out the moment of inertia, let's first start by trying to sketch a diagram of our wooden plank and its axis that is parallel to the length. So let's say that this is our plank with a length of 180 centimeters and a width of 75 centimeters. OK. Next, let's put our axis in that's parallel to its length. So let's say that this is our axis in blue here. And then let's try to represent the center of our plank with this dotted black line. OK. Now, in our problem. We're told that the axis is 10 centimeters away from one edge of our rectangle of our wooden plank. OK. Our rectangular wooden plank. So here we can say that this distance, this dimension is 10 centimeters. OK. And our center, the center of our plank is going to be a half of 75 centimeters. OK? Which is going to be 37.5 centimeters. And let me just make some space here in this dimension for that number. OK. And no, from our diagram, I think this is a pretty good diagram demonstrating what we have here. We're trying to figure out a moment of inertia with our axis parallel to the length of the wooden plank inside the plank. Now, what's that moment of inertia going to be? Well, our moment of inertia which we can call a total is going to be the moment of inertia for the rectangular object about its axis plus the moment of inertia about about the axis that's parallel to its length. So if we can find both of those, then we should be able to find our total moment of inertia. Now let me put that in red here. What do we know about both of those? Well, recall, OK, that for the rectangular object about an axis that's parallel to its length and located at the center of mass, then its moment of inertia is going to be 1/12 of M multiplied by its width squared. OK. And for our axis parallel to its length, OK, then its moment of inertia is going to be equal to its mass multiplied by the distance that it's away from the center of mass squared. And we can call that D, so that means the total moment of inertia is going to be 1/12 of MW squared plus MD squared. Now let me just refer to both of these here. OK. Uh Because notice that we have, we have most of our information, we know the mass of the plank, we know what its width is, but we don't know the distance yet that our axis is from our center of mass, but we can figure it out if we look on our diagram. Remember we said it's 10 centimeters away from one edge. Our center of mass or the center of our plank is at 37.5 centimeter mark. So that means the distance D is going to be different be the difference between those two dimensions, 37.5 minus 10, which is going to be 27.5 centimeters. So we have a value for ad know that we have all this information. We can go ahead and talk because no, that means it's going to be 1/12. Awesome. Well, we could even factor over to our mass here. Let me do that first. So it's going to be the mass 1/12 of W squared plus D squared. And no substitute in our values. We know our mass is 15 kg. OK. The width of the plank is 75 centimeters. So we can change that to meters, which is gonna be 0.75 m. So that's going to be squared plus the distance away from the distance of the axis away from the center mass D which is 27.5 or 0.275 m squared. That's how we can find our total moment of inertia. Now, when we go ahead and substitute those values, then we should get it to be equal to 1.8375 kg square meters. And if we approximate that to or if we write that to two significant figures, it will be approximately 1.8 kg square meters. Therefore, that tells us a is the correct answer. Thanks a lot for watching everyone. I hope this video helped.