Moment of inertia or inertia of any body is defined as a measure of the resistance of a body to angular acceleration about a given axis of rotation. Moment of inertia is calculated as the sum of the products of each element of mass of a body and the square of the elements’ distance from the axis of rotation. This article will tell you how to calculate inertia in simple steps.
Moment of inertia of a body in angular motion is represented as “I”. The moment of inertia of a body not only depends upon the mass and shape of a body but also depends upon the distance of the axis of rotation.
Formulae for calculating moment of inertia
- In case of a point mass the moment of inertia is the product of mass and square of the radius from the axis of rotation.
This is represented as:
I = mr2
- In case of an object with an axis of symmetry, it is some fraction of that which it would have if all the mass is at the radius.
This is represented as:
I = kmr2
- In case of a number of point masses revolving around a common axis of rotation, placed at different location, moment of inertia is the summation of individual moment of inertia.
This is represented as:
I =∑ miri2 = m1r12 + m2r22 + m3r32 +…..
- In case of a continuous mass distribution the moment of inertia is an infinite sum of all the point mass moments which make up the whole mass.
This is represented as:
I = = dm
The mass of the element depends upon the geometry or shape of the body. This formula can be used along the principal axis of symmetry of objects. If you want to use moment of inertia of the objects along the arbitrary axis then the formula is different.
The moment of inertia of any body is largely influenced by the distance of axis of rotation or the position of axis of rotation from its centre of mass. Some of the formulas for calculating moment of inertia around a cylinder, hoop and a rod around various axis of rotation are given below.
- Moment of inertia of a solid cylinder
Around its symmetrical axis
I = 1 MR2
2
Around the central diameter
I = 1 MR2 + 1 ML2
4 12
- Moment of inertia around a hoop
Around its symmetrical axis
I = MR2
About its diameter
I = 1 MR2
2
- Moment of inertia of a rod
About its center
I = 1 ML2
12
About its end
I = 1 ML2
3
Moment of inertia is used in the calculation of a large number of parameters in the angular motion of a body. It is used in the calculations of force (torque in angular motion), kinetic energy, momentum, and work energy of a body performing angular motion. The mass of a body in linear motion is replaced by the moment of inertia in the equations of angular motion.
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