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Tuesday, July 14, 2015

Moment of inertia

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The Moment of Inertia or Mass Moment of Inertia is the measure of a body's resistance to change in it's rotational speed. The moment of inertia must be specified with respect to a chosen rotational axis.
The moment of inertia depends on the body's mass distribution and the rotational axis chosen. The larger moment of inertia requiring more torque to change the body's rotational speed.

A point mass
The moment of inertia is the mass times the radius from the rotational axis squared.
Moment of Inertia of Point Mass
A collection of point mass
The moment of inertia is just the sum of the point mass moment of inertia.
Moment of Inertia of Collection of Point Mass
A continuous mass distributions 
The moment of inertia require an infinite sum of all the point mass moment of inertia which make up the whole part. This can be calculated by an integration over the whole mass.
Moment of Inertia of Continuous Mass Distributions
A common shape
For a common shape, we usually calculate the moment of inertia about the certain point and use it to apply for another rotation axis by use the parallel axis theorem.
Moment of Inertia of Common Shape Body
Parallel Axis Theorem
The moment of inertia about any rotation axis which parallel to a certain axis at center of mass (Iparallel)  is the moment of inertia about the center of mass (Icm) plus the product of mass times the distance between the center of mass and the rotation axis squared.
Parallel Axis Theorem
Parallel Axis Theorem can also apply to any rotation axis which parallel to  any certain axis at a point that already know the standard moment of inertia (Ixx , Iyy or Izz).
Parallel Axis Theorem & Radius of Gyration

Radius of gyration [K]
It is the distance from a rotation axis to a certain point which the mass of a body may be assumed to be concentrated and at which the moment of inertia will be equal to the moment of inertia of the actual mass about the rotation axis.

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