Lecture #16

Reading: A:4

Moments of Inertia for Rotated Axes

Consider an area in a coordinate system x,y which is rotated an angle q to x’,y’ .

If we want to find the moments of inertia with respect to the x’,y’ axes, we could integrate.

I_x’ = ∫ ( y’ )² dA I_y’ = ∫ ( x’ )² dA I_x’y’ = ∫ ( x’ )( y’ ) dA

These integrals can be written in terms of the x,y coordinate system if we relate x’ and y’ to x and y. Note that

x’ = x cos q + y sin q y’ = y cos q - x sin q

Now, we can substitute these equations into our integrals. Consider I_x’ .

I_x’ = ∫ ( y’ )² dA = ∫ ( y cos q - x sin q )² dA

I_x’ = ∫ ( y² cos² q + x² sin² q - 2 x y sin q cos q ) dA

Now, we can separate the integral into three integrals. Recognizing that q is a constant with respect to the integration, we get

I_x’ = ∫ y² dA cos² q + ∫ x² dA sin² q - 2 ∫ x y dA sin q cos q

Recognizing the definitions of I_x , I_y and I_xy ,

I_x’ = I_x cos² q + I_y sin² q - I_xy 2 sin q cos q