∵CE The object shown in FlGURE 10-38 can be rotated in three...

$\because$ CE The object shown in FlGURE $10-38$ can be rotated in three different ways: case $1,$ rotation about the $x$ axis, case $2,$ rotation about the $y$ axis; and case $3,$ rotation about the $z$ axis. Rank these three cases in order of increasing moment of inertia. Indicate ties where appropriate.

$\because$ CE The object shown in FlGURE $10-38$ can be rotated in three different ways: case $1,$ rotation about the $x$ axis, case $2,$ rotation about
the $y$ axis; and case $3,$ rotation about the $z$ axis. Rank these three
cases in order of increasing moment of inertia. Indicate ties where
appropriate.

Key Concepts

Moment of Inertia

A physical quantity that represents how difficult it is to change the rotational motion of an object, determined by both the mass of the object and how that mass is distributed relative to the axis of rotation.

Axis of Rotation

A specific line in space about which an object rotates, with the moment of inertia varying depending on the orientation of this axis relative to the object’s mass distribution.

Mass Distribution

The manner in which an object's mass is spread out in space; areas of the object that are farther from the axis of rotation contribute more significantly to the overall moment of inertia, affecting the ease or difficulty of rotational motion.

Transcript

0:00 Hello everyone.

00:01 This is problem 73 from chapter 10.

00:04 Says the object shown in the figure can be rotated in three different ways.

00:08 Case 1 is rotation about the x -axis, case 2 about the y and case 3 about the z -axis.

00:12 And so as i rank these three cases in order of increasing moment of inertia indicate ties were appropriate.

00:19 So i've drawn a very rough sketch of the figure shown.

00:23 So along the y -axis, which is this one here, we have two balls of mass big m over 2.

00:29 I guess everything's big m, so i'll say m over 2, separated from the origin by 2r.

00:39 And then along the x -axis, we have balls mass m, r away from the origin, and on the z -axis we have two balls of mass 3m, which are positioned r away from the origin.

00:54 So the simple thing to do in this case is to just go about finding the moments of inertia.

01:02 So i'll just call these i, x, i, y, and i z.

01:05 So rotation about the x -axis, remember, remember that moment of inertia is just i equals the sum over little i, m sub i, r sub i squared.

01:19 Okay, so in order to find it about the x -axis, we know that these two contribute nothing because we're rotating about the x -axis, but then we have a contribution from those positioned along the y and those positioned along the z -axis.

01:34 So, first do, first we know we're multiplying by two here, because there are going to be two of these things...

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