Theorems of Pappus (Finding mass)

[Reefy]

Homework Statement

A steel die, shown in section, has the form of a solid
generated by revolving the shaded area around the
z-axis. Calculate the mass m of the die.

Homework Equations

V =θ(rA) where r is the distance from the origin to the centroid of each shape

m = ρV where ρ = 7830 kg/m³

The Attempt at a Solution

Obviously θ = 2π.

If I make rA = ( r₁A₁ - r₂A₂ ), then A₁ = area of right most rectangle = (100 mm x 200 mm) and r₁ = distance from origin to that rectangle's centroid = 60 mm + ( 100 mm / 2 ) = 110 mm.

Also, A₂ = area of half-circle = π(R²)/2 = π(60²)/2 ≅ 5654 mm and r₁ = distance from origin to that half circle = 60mm + 4R/(3π) ≅ 85.46 mm.

This method gives me correct volume (according to answer given) which in turn gives me correct mass m ≅ 84.5 kg. However, how come I don't have to take into account the rectangle of area (60mm x 200mm) (which is to the left of the half circle) along with its coordinate from origin to centroid (which would be 30mm)? Wouldn't I have to make that r₃A₃ and subtract that from ( r₁A₁ - r₂A₂ ) ?

 

[NascentOxygen]

how come I don't have to take into account the rectangle of area (60mm x 200mm) (which is to the left of the half circle)

Because that area is part of the void, it's empty. The question asks you to calculate that volume shown cross-hatched.

 

[Reefy]

Because that area is part of the void, it's empty. The question asks you to calculate that volume shown cross-hatched.

By "shaded area", they mean the cross-hatched section right? Wouldn't that make the half circle area void as well?

edit: never mind I get what you're saying. Thanks lol. The volume of only the cross hatched section means obtaining that area and coordinate only. Which means I have to subtract the half circle area and its coordinate. OK, I guess I thought the problem wanted the whole figure not just the cross hatched section. Thanks again