Moment of Inertia of Disc through diameter

[unscientific]

Homework Statement

The problem is attached in the picture.

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.

 

[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.

 

[unscientific]

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.

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?

 

[voko]

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

 

[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 dI_x = y²dm, to the MOI around the y-axis as dI_y = x²dm, and to the MOI around the z axis as dI_z = (x²+y²)dm = x²dm + y²dm, which, using the two previous equations, means that dI_z = dI_y + dI_x (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, I_z = I_x + I_y.

 

[unscientific]

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

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