Parallel Axis Theorem problem help

[LeFerret]

Homework Statement

This is moreof a conceptual question regarding the parallel axis theorem and area.
Lets say I have a solid rectangle with length L, and Width W.
I cut out a circle of radius R at the center.
When calculating the Total Area moment of inertia of the hollow region where the circle was cut out
is it I_total=I_rectangle-(I_central-Ad²) or
I_total=I_rectangle-(I_central+Ad²)
because I know the region cut out has a negative area, so I subtract that moment of inertia from the rectangle's moment of inertia, but I'm not sure if that negative carries over inside the parenthesis as well

Homework Equations

I_any=I_central + Ad²

The Attempt at a Solution

 

[SteamKing]

If you have a rectangle and then you remove a circular area from the center, the parallel axis theorem really doesn't come into play, because the centroid of the resulting figure hasn't changed from before the circular area was removed; i.e. I(net) = I(rectangle) - I(circle)

 

[LeFerret]

I agree, but I wanted to explain what I was asking as simply as possible, the circle could be removed anywhere and it could be any series of shapes attached together. What I am confused on is if that negative area comes to play inside of the parenthesis or if it's |Ad^2|

 

[ehild]

Homework Statement

This is moreof a conceptual question regarding the parallel axis theorem and area.
Lets say I have a solid rectangle with length L, and Width W.
I cut out a circle of radius R at the center.
When calculating the Total Area moment of inertia of the hollow region where the circle was cut out
is it [STRIKE]I_total=I_rectangle-(I_central-Ad²)[/STRIKE] or
I_total=I_rectangle-(I_central+Ad²)
because I know the region cut out has a negative area,

You cut out a positive area, that makes a negative contribution to the total moment of inertia.

so I subtract that moment of inertia from the rectangle's moment of inertia, but I'm not sure if that negative carries over inside the parenthesis as well

Homework Equations

I_any=I_central + Ad²

The Attempt at a Solution

The second equation is correct.

ehild

 

[LeFerret]

thank you!

 

[ehild]

You are welcome

ehild