# Theorems of Pappus

1. Nov 17, 2014

### _N3WTON_

1. The problem statement, all variables and given/known data
The lampshade shown is constructed of $0.6 \hspace{1 mm} mm$ thick steel and is symmetric about the z-axis. Both the upper and lower ends are open. Determine the mass of the lampshade. Take the radii to be to midthickness.

2. Relevant equations

3. The attempt at a solution

Last edited: Nov 17, 2014
2. Nov 17, 2014

### _N3WTON_

Sorry about the image...I'm trying to figure out how to fix it :(

3. Nov 17, 2014

### _N3WTON_

Obviously my answer is wayyy off (unless this is the worlds heaviest lampshade) but I'm not entirely sure where I'm going wrong..

4. Nov 17, 2014

### Staff: Mentor

I haven't checked all of your numbers, but there's one thing that stands out. 42,181 mm2 ≠ 42.181 m2. You're way off (by a factor of 1000) here.

5. Nov 17, 2014

### _N3WTON_

Wow thank you, so after editing the calculation I get:
$42000 \hspace{1 mm} mm \hspace{1 mm} / 1000000 = 0.042 \hspace{1 mm} m$
$(0.042m^{2})(7830\frac{kg}{m^{3}})(6x10^{-4}m) = 0.197 \hspace{1 mm} kg$ This answer makes a whole lot more sense, but I am still off by about .1 kg, the answer given is: $0.293\hspace{1 mm} kg$. Is there something I need to do with the radii of the shade? I have a hunch that when they mention the radii to be midthickness that I should account for that somehow, but I'm not entirely sure how...

6. Nov 17, 2014

### SteamKing

Staff Emeritus
If this lampshade is a body of revolution, like most lampshades, you have seriously misinterpreted the Pappus Centroid Theorem in order to calculate the area of the surface of revolution:

http://en.wikipedia.org/wiki/Pappus's_centroid_theorem

Since the shade is fabricated from steel with a thickness of 0.6 mm, you want to calculate the arclength of the shape of the shade and figure out its centroid from the z-axis, not what you have done.

You should pretend the shade is laid out as 3 skinny rectangles, with each rectangle having a width equal to the thickness of the steel, 0.6 mm. IOW, the top and bottom of the shade are cylinders and the middle portion is the frustum of a cone.

The centroid of each segment is going to lie somewhere inside the corresponding skinny rectangle, which is why the OP gives the hint, "Take the radii to be to midthickness."

Last edited: Nov 18, 2014
7. Nov 17, 2014

### _N3WTON_

Ok I believe I have solved it (or I came to the correct answer by mistake). I didn't exactly use the method prescribed by SteamKing because I was having trouble setting up the proper equations, this is what I did instead:

8. Nov 18, 2014

### SteamKing

Staff Emeritus
There's almost no equations to set up. You're dealing with straight line segments. And yes, the answer you got was fortunately close to the correct one. The next time, you may not be so lucky.

All you have to do is calculate the lengths of three lines. The lengths of the top and bottom lines can be determined by inspection. The length of the middle line requires the use of the Pythagorean Theorem.

Similarly, the distance of the centroid of each of the three lines can be easily determined. The centroids of the top and bottom lines are equal to the radii measured from the z-axis; the centroid of the middle line requires that you be able to calculate the mean radius of the endpoints of this line.

Once you have the basic data determined, you can find the total length of the three lines and their centroid and then apply the Theorem of Pappus to find the volume of metal required to fabricate the lampshade.

9. Nov 18, 2014

### _N3WTON_

Ok, so the problem is basically a calc 1 problem (rotating surface)? Also, just to be clear you're saying the method I used is incorrect and that I found the correct answer by coincidence? I'm going to try the problem and I'll post my results, thanks for the help