Moment of Inertia of Discrete Particles
An object consists of several point particles. Such particles are known as discrete particles. If a discrete particle rotates about a certain axis, how do you calculate the value of the moment of inertia?
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00:00An object is composed of a number of point particles.
00:06All these point particles constitute a single object.
00:12This means that if one point particle moves, all the other point particles will also move together.
00:22There is no difference in distance between point particles.
00:28In other words, the shape of the object is fixed.
00:34If an object like this rotates about a certain axis, what is the value of its moment of inertia?
00:46Let's assume the object is located on a two-dimensional surface, or XY plane.
00:51And, the axis of rotation is the Z-axis.
00:57Each point particle will take its own path.
01:03However, the angular velocity of each particle is the same.
01:09That's why the shape of the object doesn't change.
01:21In cases like this, the value of the moment of inertia of the object is the sum of the moments of inertia of each point particle that makes it up.
01:33For identification, let's call them Particle 1, Particle 2, and Particle 3.
01:41Since there are three point particles, and their axis is the Z-axis, we can write, I sub Z equals I1, plus I2, plus I3.
01:54For Particle 1, the mass is m1, and the distance to the axis is r1.
02:00Then the moment of inertia of Particle 1 is m1r1 squared.
02:09For Particle 2, the mass is m2, and the distance to the axis is r2.
02:15Then the moment of inertia of Particle 2 is m2r2 squared.
02:23For Particle 3, the mass is m3, and the distance to the shaft is r3.
02:29Then the moment of inertia of Particle 3 is m3r3 squared.
02:37Regularities like this can be written in sigma notation.
02:43This is the moment of inertia of an object composed of several point particles.
02:51Hopefully it's useful.
02:53And, don't forget to watch the next video.