A point particle, a particle that has mass but no size, rotates about a fixed axis. How do we calculate the moment of inertia of a point particle.
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00:00An object rotates through a specific axis.
00:07In the previous series, we were familiar with the term moment of inertia,
00:11which is nothing but a measure of an object's ability to maintain its angular velocity.
00:19This quantity is often symbolized as big I. I is for inertia.
00:24The moment of inertia is proportional to the mass of the object.
00:33The moment of inertia is proportional to the radius of the object relative to the axis.
00:39Mathematically, the moment of inertia can be calculated through the equation I equals m r
00:43squared. This equation is valid for point particles. An object that has mass but no size.
00:54To understand this equation further, in a three-dimensional Cartesian coordinate system,
01:05there are point particles with coordinates r sub x, r sub y, r sub z, where the r sub z value is zero.
01:14We will calculate the value of the moment of inertia about three axes, the x-axis, y-axis,
01:20and z-axis. So, we will duplicate this system into three.
01:31In the first system, the particles will rotate about the x-axis.
01:36This means that the axis is the x-axis.
01:41We should be able to imagine this particle rotating about the x-axis.
01:45Even though the intersection point on the x-axis is r sub x,
01:48the distance from the point to the axis is r sub y.
01:54Thus, the value of the moment of inertia on the x-axis, I sub x is equal to m r sub x squared.
02:03In the second system, the particles will rotate about the y-axis.
02:09This means the axis is the y-axis.
02:12This means the axis is the y-axis.
02:17Now the distance of the point to the axis is r sub x.
02:22Thus, the value of the moment of inertia on the y-axis, I sub y is equal to m r sub y squared.
02:32In the third system, the particles will rotate about the z-axis.
02:37This means that the axis is the z-axis.
02:42The distance from the point to the axis is the length of the line perpendicular to the axis leading to the point particle.
02:50Using the Pythagorean theorem, the length of this line is the root of r sub x squared, plus r sub y squared.
02:59Thus, the value of the moment of inertia on the z-axis, I sub z is equal to m,
03:04multiplied by r sub x squared plus r sub y squared.
03:12From here, we can see that the same particle can have several values of moment of inertia depending on its axis.
03:23Thank you for watching this video.
03:26And, don't forget to watch the next video.