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Strategy in Calculating Mass Moment of Inertia

In a Nut Shell: Integration provides a method to calculate mass moment of inertia.

Recall Definition of Mass Moment of Inertia about an Arbitrary Point, A, I^A_z,K

I^A_z,K = ∫ ∫ ρ r² dx dy = = ∫ ∫ r² dm

where I^A_z,K is the mass moment of inertia of the body about an arbitrary point, A

ρ is the mass per unit area of the body (mass density)

r is the distance from the arbitrary point, A, to the element of mass, dm

dm equals ρ dx dy

dx dy is the element of area of the body

See the figure below.

Strategy: 1. Show the element of area, dx dy and its location at (x,y).

2. Calculate the element of mass, dm = ρ dx dy where ρ = ρ(x,y)

3. Calculate r² : r² = (x ˗ x_A)² + (y ˗y_A)²

4. Set up integral: ∫ ∫ ρ(x,y) r² dx dy = ∫ ∫ ρ(x,y) [(x ˗ x_A)² + (y ˗y_A)² ] dx dy

5. Determine limits of integration for x and for y.

6. Evaluate the integral.