Theorems of Pappus (Finding mass)

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In summary, the problem is asking for the mass of a steel die that is generated by revolving a shaded cross-hatched area around the z-axis. The volume and mass can be calculated using the equations V = θ(rA) and m = ρV, respectively. The correct solution is obtained by only considering the cross-hatched section and its coordinate, as the other sections are part of the void.
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Reefy
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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.

64dt2d.png

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 ) ?
 
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  • #2
Reefy said:
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.
 
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NascentOxygen said:
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
 

1. What is the Theorem of Pappus?

The Theorem of Pappus is a mathematical principle that states that the volume of a solid generated by a plane curve rotating around an axis outside the curve is equal to the product of the curve's length and the distance traveled by its centroid.

2. How is the Theorem of Pappus used to find mass?

The Theorem of Pappus can be used to find the mass of an object by first finding the volume of the object using the theorem. Then, the mass can be calculated by multiplying the volume by the object's density.

3. What is the centroid mentioned in the Theorem of Pappus?

The centroid is the point at which all the mass of an object is evenly distributed. In the context of the Theorem of Pappus, it is the point at which the object would balance on a fulcrum.

4. Can the Theorem of Pappus be applied to any shape?

Yes, the Theorem of Pappus can be applied to any shape as long as it follows the principle of rotation around an axis outside the curve.

5. Are there any practical applications of the Theorem of Pappus?

Yes, the Theorem of Pappus has many practical applications in fields such as engineering and architecture. It is used to calculate the mass and volume of various objects and structures, including pipes, gears, and even buildings.

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