Moment of Inertia of a Point Particle

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.

Transcript

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.