Tuesday, 7 March 2017

DERIVATION OF MASS MOMENT OF INERTIA

DERIVATION OF MASS MOMENT OF INERTIA


Now we are going further to start our discussion to understand the basic concept of mass moment of inertia with the help of this post.

 

Let us first see here the basic concept of mass moment of inertia

Mass moment of inertia is basically defined as the sum of second moment of mass of individual sections about an axis.

Let us see the following figure which indicates one lamina with entire mass M. Let us assume that lamina, displayed here, is made with number of small elemental masses m1, m2, m3, m4 …etc. 
As we have considered above that the total mass of the lamina is M, now we need to determine here the mass moment of inertia for this lamina.

Where,
x1= Distance of the C.G of the mass m1 from OY axis
x2= Distance of the C.G of the mass m2 from OY axis
x3= Distance of the C.G of the mass m3 from OY axis
x4= Distance of the C.G of the mass m4 from OY axis

Similarly, we will have
y1= Distance of the C.G of the mass m1 from OX axis
y2= Distance of the C.G of the mass m2 from OX axis
y3= Distance of the C.G of the mass m3 from OX axis
y4= Distance of the C.G of the mass m4 from OX axis

Total mass of the lamina = Sum of all small elemental masses
A = m+ m+ m+ m+ ----------

Moment of mass about the OY axis will be determined by multiplying the mass with the perpendicular distance between the C.G of mass and axis OY.

Moment of mass about the OY axis = M .X

Let us determine the moments of all small elemental masses about the OY axis and we will have following equation.

Moments of all small elemental masses about the OY axis = m1.x+ m2.x2 + m3.x3+ m4.x+…….

Above equation will be termed as the first moment of mass about the OY axis and it is used to determine the centroid or centre of gravity of the mass of the lamina.

If the first moment of mass will again multiplied with the vertical distance between the C.G of the mass and axis OY, then we will have second moment of mass i.e. M.X2.

 
By taking the definition of mass moment of inertia in to consideration, we can write the equation for the sum of second moment of mass of all small elemental masses about the OY axis and we will have following equation.

Second moments of all small elemental masses about the OY axis = m1.x12 + m2.x22 + m3.x32+ m4.x42 +...
(Im) yy = m1.x12 + m2.x22 + m3.x32+ m4.x42 +…….
(Im) yy = Σ m.x2

Similarly, we will determine the sum of second moment of mass of all small elemental masses about the OX axis and we will have following equation.

(Im) xx = m1.y12 + m2.y22 + m3.y32+ m4.y42 +…….
(Im) xx = Σ m.y2

Therefore, we can say here that mass moment of inertia about an axis could be calculated by taking the product of mass and the square of the perpendicular distance between the center of gravity of the mass and that axis.

 
Therefore, we have following formula for determining the mass moment of inertia around the X axis and Y axis and it is as mentioned here.

Ixx = Σ m.y2Iyy = Σ m.x2

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 important topic i.e. moment of inertia of area under curve, in the category of strength of material, in our next post.

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