Calculate the moment of inertia of the array of point objects shown in Fig. 8–47 about

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. The moment of inertia about the y-axis is the sum of the moment of inertia's of each of these four point masses. Each of them has a moment of inertia formula equal to I equals m times its distance to the axis of rotation squared. So we have one little mass and one large mass, both of them are distance of 0.50 meters away from the y-axis. And so we go m plus M times 0.50 meter squared and then add to that the moment of inertia's for each of these two distant masses here and so that's little m plus M times its 1 meter distance away from the y-axis squared. And so substituting in values for m and M: we have 2.2 kilograms plus 3.4 kilograms times 0.50 meters squared plus 2.2 plus 3.4 times 1.00 meter squared which gives 7.0 kilogram meter squared moment of inertia about the y-axis. Now for the x-axis, we have everything at the same distance from the axis of rotation. So we have two little masses and and then two big masses all getting multiplied by the same thing; this separation here divided by 2 because we are told that the x-axis cuts down the middle and so the distance from this axis of rotation to any of the masses is 0.5 meters divided by 2 and so we square this denominator which makes 4 and then that cancels with a... this a common factor 2 on top and bottom in other words and we end up with just divide by 2 in the end after squaring the denominator. And so we have m plus M times 0.50 meters squared divided by 2 and plugging in numbers you get 0.70 kilogram meter squared. So the moment of inertia is much smaller above the x-axis and we expect that since the masses are so much closer to the x-axis than they are to the y-axis. And so it is harder to accelerate the system about the y-axis since the moment of inertia is about 10 times greater about the y-axis than it is about the x-axis. So for a given angular acceleration, we would need 10 times as much torque since the moment of inertia is 10 times greater about the y-axis.