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Monday, 6 March 2017

MOMENT OF INERTIA OF UNIFORM ROD

Today we will see here the determination of moment of inertia of one uniform thin rod; we will derive here the equation to express the moment of inertia for thin rod.

Before going ahead we must have to find out few basic posts which will be related with determination of moment of inertia for various cases such as mentioned here.

Moment of inertia for rectangular section

Moment of inertia for the hollow rectangular section

Moment of inertia for circular section

Moment of inertia for the hollow circular section 

Area moment of inertia Radius of gyration 

Basic principle of complementary shear stresses

Now we will go further to start our discussion to understand the method for determining the moment of inertia of one uniform thin rod with the help of this post.

Let us consider one uniform thin rod as displayed in following figure. Let us consider one small strip of length dx and at a distance x from the YY axis as shown in figure.
L= Length of uniform thin rod
m= Mass of the uniform thin rod per unit length
M = Total mass of the uniform thin rod
M = m.L

dx= Length of small strip
x= Distance of small strip from YY axis

Now we will determine the moment of inertia of uniform thin rod about YY axis

Let us determine the mass of the small strip and we can easily write here as
Mass of the small strip = (Mass of the uniform thin rod per unit length) x (length of small strip)
Mass of the small strip = m.dx

Moment of inertia of this small strip about axis YY could be easily determined and we can write here the following equation to represent the moment of inertia of this small strip about axis YY.

Moment of inertia of small strip about axis YY = (m. dx). x2
Moment of inertia of small strip about axis YY = mx2dx
 
Now we will integrate the above equation from 0 to L in order to secure the value of moment of inertia of entire uniform thin rod about axis YY.

Finally we can conclude that moment of inertia of entire uniform thin rod about axis YY could be easily written as mentioned here.

IYY =ML2/3

Where, M = m.L
Do you have any suggestions or any amendment required in this post? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

We will see another article in our next post i.e. Determination of moment of inertia of area under curve in our next post.

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