# Moment of Inertia of Disc through diameter

1. Aug 20, 2012

### unscientific

1. The problem statement, all variables and given/known data

The problem is attached in the picture.

3. The attempt at a solution

I managed to solve it using a different method. I have no idea what the answer is talking about..

My method

Found dI of a strip = (1/3)*dm*h2

then i replace h by x, then integrate from -a to a to get the same answer.

But as for the answers, I have no clue.

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2. Aug 20, 2012

### voko

As far as I can tell, x in the answer is perpendicular to the axis against which the moment is being determined. I think you can figure out the rest.

3. Aug 20, 2012

### unscientific

The problem is, I can't fingure out what is the shape of their dM.

It looks as if they are taking a small square mass dM that is distance x away from axis (perpendicularly)

so I = x2 dM

but dM = 2σ √(a2-x2)

It looks as if √(a2-x2) is referring to the distance along the axis from the origin..But how does that make any sense? r in ∫ r2 dM refers to distance from the axis and not the origin.

Then are they using dM as a strip or a small square mass?

4. Aug 20, 2012

### voko

They are taking the entire strip parallel to the axis at distance x from it.

5. Aug 20, 2012

### Filip Larsen

To understand the general statement, it may be easier to consider only a small mass element, dm, of the disc placed at coordinate x, y (with z being zero since its a plate). This one mass element contributes to the MOI around the x axis as dIx = y2dm, to the MOI around the y axis as dIy = x2dm, and to the MOI around the z axis as dIz = (x2+y2)dm = x2dm + y2dm, which, using the two previous equations, means that dIz = dIy + dIx (make a diagram if you have trouble seing this). Since this is valid for all the mass element of the planar disc it must also be valid for the sum (or rather the integral) of all the mass elements MOI, that is, Iz = Ix + Iy.

6. Aug 20, 2012

### unscientific

Yup, I finally figured that out. Initially I worked it out by using a strip perpendicular to the axis with an MOI of dI = (1/3)x2 dM