# Moment of inertia of a leftover square

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## Homework Statement

A square plate is of mass M and length of edge 2a. Its M.I about its centre of mass axis, perpendicular to its plane is equal to I(1). Four identical disks of diameter a are cut from the plane. The MI of leftover square about the same axis?

## Homework Equations

1) MI of square plate of edge length a- Ma^2/6
2) MI of disk of radius R- MR^2/2
3) I about a point P, x distance from COM is I about COM + Mx^2

## The Attempt at a Solution

I first calculated MI of original square plate using formula 1 as (2/3)Ma^2=I(1). Mass of one disc as (pi/16)M=M(d). MI of one disc about the COM of disk=M(d)a^2/8=I(disc). MI of one disk about CM of square=I(disc)+(M(d)*a^2)/2=I(2). The MI of leftover square should be I(1)-4I(2) but I'm getting the wrong value every time, can you point out my mistake and write out the calculations in case I'm making a mistake somewhere?

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Nathanael
Homework Helper
Assuming the plate is uniform and we take the only non-overlapping configuration of circles (in each corner) then I see no mistakes in your post.

Gold Member
Assuming the plate is uniform and we take the only non-overlapping configuration of circles (in each corner) then I see no mistakes in your post.
I must be making a calculation mistake somewhere then...I'll give it a go again. Thanks a lot for the help, I'm very grateful

haruspex
Homework Helper
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Ma^2/6
The sides are length 2a, not a.

Nathanael
Homework Helper
The sides are length 2a, not a.
He prefaced that with “for edge length a.” (A bit confusing, I agree. He should’ve used an L or something.)

If you look into his calculation though, he does it right:
... using formula 1 as (2/3)Ma^2=I(1).
Every other step also seemed correct, but maybe I missed something.

haruspex
Homework Helper
Gold Member
but I'm getting the wrong value every time,
He prefaced that with “for edge length a.”
Ah yes. Thanks.

Gold Member

Ah yes. Thanks.
He prefaced that with “for edge length a.” (A bit confusing, I agree. He should’ve used an L or something.)

If you look into his calculation though, he does it right:

Every other step also seemed correct, but maybe I missed something.
Sorry, I forgot to post yesterday- since I solved it late at night. I was indeed making a silly calculation mistake over and over again but the entire method and the calculations I posted here are correct, I verified this by cross-checking my answer with the one given in my textbook. I'm sorry for wasting your time and am really grateful for your help, Thank you.