Physics with Calculus - Moment of Inertia

Question

Consider a uniform rod with length L and mass M placed on the x-axis between the points
(-1/3*L, 0) and (2/3*L, 0).
Determine the moment of inertia of the rod for rotation about the y-axis by integration.

Answer 1

λ = linear density = M/L

 

I_y = ∫x²λ(x)dx = ∫x²(M/L)dx = (M/L)∫_-L/3^2L/3 x²dx

 

I_y = [M/(3L)] (x³)_-L/3^2L/3 = [M/(3L)] [8L³/27 - (-L³/27)]

 

I_y = [M/(3L)] L³ (9/27) = (ML²/3)(1/3) = ML²/9

 

Megan, note this gives the same answer as using the parallel axis theorem, which Steven W. suggested at

https://www.wyzant.com/resources/answers/258089/center_of_mass_and_moment_of_inertia_of_extended_objects