Question

find moment of inertia of semicircular disc about Axis passing through centre of mass and perpendicular to its plane

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Answer 1

Consider a semicircular plate with radius ( R) and thickness ( t) rotating about the Z-axis which is perpendicular to the plate (out of the page at Z):

Mass moment of inertia is defined as:

I=mr2

m= mass

r= perpendicular distance between the mass and the axis of rotation.

For a complex shape like this semi-circular plate, the mass varies with radius so we define the moment of inertia as:

I=∫r2dm

I begin by defining an infinitely narrow curved rectangular-shaped element with curved length πr and width dr and area = dA = (πr)dr as shown above (shaded). This area is distance r from the Z-axis. Integrating from r=0 to r=R will add up all the narrow curved rectangles and account for the total mass of the plate.

PART 1. First derive an expression for the mass of the curved element

mass = (density)(volume)

or

dm=ρdV=(ρt)dA

where ρ = density of the plate, t = thickness of the plate

but the area of the rectangular-shaped element dA=(πr)dr

∴ dm=ρt(πr)dr

The total mass of the semi-circular plate is:

M=ρtA=ρt(1/2)πR2

The ratio of the mass of the element to the total mass of the plate is: