Find the rotational inertia of thesystem of point particles...

Find the rotational inertia of thesystem of point particles shown in the figure assuming the system rotates about the (a) $x$ -axis, (b) $y$ -axis, (c) z-axis. The $z$ -axis is perpendicular to the $x y$ -plane and points out of the page. Point particle $A$ has a mass of $200 \mathrm{g}$ and is located at $(x, y, z)=(-3.0 \mathrm{cm}, 5.0 \mathrm{cm}, 0)$ point particle $B$ has a mass of $300 \mathrm{g}$ and is at $(6.0 \mathrm{cm}, 0$ 0), and point particle $C$ has a mass of $500 \mathrm{g}$ and is at $(-5.0 \mathrm{cm},-4.0 \mathrm{cm}, 0) .$ (d) What are the $x$ - and $y$ -coordinates of the center of mass of the system? (FIGURE CAN'T COPY)

Key Concepts

Rotational Inertia

Rotational inertia, commonly known as moment of inertia, is a measure of an object's resistance to angular acceleration about a particular axis. In the context of a system of particles, it is computed by summing the products of each particle's mass with the square of its perpendicular distance from the rotation axis.

Axis of Rotation

The axis about which a system rotates directly affects the calculation of moment of inertia. Changing the axis (for example, the x-axis, y-axis, or an axis perpendicular to the plane) changes the perpendicular distances for each mass, thereby influencing the overall rotational inertia.

Point Particle Systems

When dealing with systems of point particles, each particle is treated individually in calculations. The total moment of inertia is computed as a sum over the contributions from each particle, where each contribution is the product of the particle’s mass and the square of its distance from the axis of rotation.

Center of Mass Calculation

The center of mass of a system of particles is the point at which the weighted positions of the masses balance out. It is calculated by taking the average of the positions of the masses weighted by their individual masses, providing a simplified description of the system's distribution of mass.

Transcript

00:01 So in this problem, we want to figure out the moment of inertia of a system with respect to three different axes, the x, y, and z axes.

00:08 So the setup is we have the x, y, plane here, and you can imagine the z axis kind of coming out of the page.

00:16 And the three masses are mass a, b, and c.

00:22 Mass a has a mass of 200 grams, mass b has a mass of 300 grams, and mass c has a mass of 500 grams.

00:31 Okay, a is located at the position minus 3, 5, where the coordinates are given in centimeters.

00:38 Mass b has a location of 6 -0, and mass c is the location of minus 5, minus 4.

00:44 So they're all in the x -y plane, so none of them have z components.

00:50 Okay, so first we want to figure out the moment of inertia of this system if we were to rotate about the x -axis.

00:57 So around this x -axis here.

01:00 Okay, so the definition of moment of inertia is the sum of each mass times its position squared.

01:09 And my position, i mean distance from axis of rotation.

01:14 Okay, so here, if we start with mass a, our first term in the sum will be, well, mass a has a mass of 200 grams, so if we convert that to kilograms, it'll be 0 .2 kilograms times its distance from the axis.

01:31 Of rotation.

01:33 So how far is point a from the x -axis? well here the distance a from the the distance that a is from the x -axis, that's just the y component of a, which in this case is 5 centimeters or 0 .05 meters.

01:56 So the first term is 0 .2 kilograms times 0 .05 meters squared.

02:04 So now we want to do the same thing for mass b.

02:07 So mass b, well, it has a mass of 300 grams.

02:11 So if we convert that to kilograms, that's just 0 .3 kilograms, then times how far it is from the x -axis.

02:20 Well, mass b is on the x -axis, so it's a distance of 0 from the x -axis, so it doesn't contribute to the moment of inertia.

02:29 And lastly, mass c is going to have a mass of 0 .5 kilograms, and again, its distance from the x -axis is given by its y component, in this case negative 4 centimeters.

02:44 So if we convert that to meters, our last term becomes 0 .5 times minus 0 .04 squared.

02:54 And if you plug all of that into your calculator, it should turn out to be, running out a space here, 0 .0013.

03:05 3 and the units will be kilograms meters squared.

03:12 Okay, so let me scroll down here.

03:16 Now we want to do the same thing for the y -axis.

03:25 So if we want to rotate about the y -axis then the only thing that's going to change is now the distance, the distances will be different.

03:40 The r sub -i's will be different because now we're concerned with how far we are from the y -axis, not the x -axis.

03:55 Okay, so starting again with mass a, how far is mass a from the y -axis now? well, that is just the x of a, which is minus three centimeters.

04:14 So our first term, again, will be 0 .2 times, if we convert to meters, minus 0 .0 .0 .0.

04:24 0 .03 squared.

04:30 And now for mass b, how far is mass b from the y axis? well, it's a distance of 6 from the y axis...

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