Centroid of a composite area

1. Nov 5, 2015

goldfish9776

1. The problem statement, all variables and given/known data
i have a few question here .
1. the y bar for I should be 16.98, am i right ?
2. The y bar for II should be 50+12+24=86 , am i right ?
3. the x bar for V should be 50 , am i right ?

2. Relevant equations

3. The attempt at a solution

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2. Nov 5, 2015

haruspex

The formula $\bar y = \frac{4r}{3\pi}$ is a general formula for the distance of the centroid of a semicircle from its centre of radius. In the coordinates used in the problem, that centre is not the origin.

3. Nov 6, 2015

goldfish9776

ok, the y bar of II should be 74 , right ? (50+12+12)

4. Nov 6, 2015

haruspex

It's not clear to me exactly what the 50 in the diagram refers to. I have to agree that it looks more like it is to the low point of II, not to its centre. Is its radius 12? Where is that indicated?

5. Nov 6, 2015

goldfish9776

it's not stated . Anyway , if we notice the calculation of the volume in the first photo , the d = 48 , so r = 24 ? so , the y bar for II should be 50+12+24 = 86 ?

6. Nov 6, 2015

SteamKing

Staff Emeritus
As far as the radius of hole II is indicated, look at Hole V on the diagram. Notice that a radius arrow is drawn from the center of Hole V with the notation "24 Two Places" beside it. This is drafting shorthand for "the radius of this hole is 24 (mm) and the radius of the other hole is also 24 (mm)." This indicates that the radius of Hole II is identical to the radius of Hole V.

7. Nov 6, 2015

SteamKing

Staff Emeritus
I think the 50 dimension on the vertical piece indicates where the point of tangency between the top radius and the sides is located. The center of Hole II is assumed to be located in line vertically with this location unless otherwise indicated on the drawing. Notice how the center of Hole V on the bottom of the piece has dimensions which show it offset 40 mm from the x-axis and back 45 mm from the edge of the piece.

8. Nov 6, 2015

haruspex

Ok, so you agree with the y coordinate of the centroid of II being 86, as deduced in post #5? It looks right to me.

9. Nov 6, 2015

SteamKing

Staff Emeritus
No, I don't.

The bottom piece IV is 12 mm thick. The center of hole II is located 50 mm above the top of IV, so that locates the center of IV = 50 + 12 = 62 mm above the xz-plane.

For some reason, the OP thinks the centroid of hole II is located 24 mm above that, which would put it at the top of the hole, which is clearly incorrect. IOW, the OP's location would put the centroid of hole II higher than the centroid of piece I.

10. Nov 6, 2015

haruspex

The diagram is drawn as the view from infinity, so parallel lines appear parallel. If you take a line parallel to the z axis through the point on the right marked at 50mm above IV it quite clearly passes through the lowest point of II, not its centre.

11. Nov 6, 2015

SteamKing

Staff Emeritus
Respectfully, I disagree. I do note that someone has sketched by hand a light line roughly tangent to the bottom of hole II, but the end of this sketched line intersects the edge of the vertical piece below the end of the 50 mm vertical dimension.

As I mentioned previously, normal drafting practice is to show dimensioned locations for the centers of any circles which are not concentric with one another. I still maintain that hole II is concentric with the radius which forms the top of piece I, and that the center of both circles is located 50 mm above the top of piece IV.

Besides, the table in DSC_0171.jpg gives the locations of all the centroids in this figure, and these seem accurate to me. It's going to take indisputable visual evidence to overturn these locations, IMO.

12. Nov 6, 2015

haruspex

Then we will have to agree to disagree.
The hand drawn line is clearly not parallel to the base, maybe because it is drawn in a natural perspective. If you draw it accurately it aligns with the top of the 50mm mark. Moreover, II does not look anything like concentric with IV.
The accompanying table I take as being the answer set, not as side information. I remain convinced the 62mm either is an error in itself or indicates an error in the drafting of the diagram.

13. Nov 6, 2015

SteamKing

Staff Emeritus
Well, I would hope that the answer set would not contain any erroneous information. That defeats the purpose of having an answer set.

14. Nov 6, 2015

goldfish9776

the line is tangential to the circle is drawn by me ... it's to show the separation of I and III . why the y bar for II isnt at the center of the circle , but rather on the tangent of the circle ?

15. Nov 6, 2015

SteamKing

Staff Emeritus
y-bar for hole II is at the center of hole II. It's not clear which tangent you are talking about.

The table gives y-bar for II as 62 mm above the xz-plane, which is 12 mm for the thickness of piece IV plus the 50 mm vertical distance, which is where upper radius of piece I intersects with the sides of piece III.

16. Nov 6, 2015

haruspex

It comes down to how one reads the alignments off the diagram. goldfish and I read it one way, you and the results table read it another.
In the attached, I've added a construction line to make the relationship between the 50mm measure and piece II clearer. I have made it accurately parallel to the baseline, though the hand-sketched line might make you think it isn't.

17. Nov 6, 2015

SteamKing

Staff Emeritus
I still think you're pinning your hopes on a poorly drafted figure, which may or may not be distorted by all the copying going on.

If you look at the most recent image, it would appear that hole V is located closer to the edge of piece IV in the direction of the positive z-axis, yet according to the dimensions of the piece, hole V is supposed to be located 40 mm from either side of piece IV.

But, for the sake of argument, let's take your line as being located at 50 + 12 = 62 mm above the base of piece IV. Since piece IV is 40 + 40 = 80 mm wide, this means that the radius of the top of piece I is 40 mm. This would put the top of piece I at 12 + 50 + 40 = 102 mm above the base of piece IV. Now, assuming that the bottom of hole II is tangent to the line which is 62 mm above the base of piece IV, and also using the fact that hole II has a diameter of 24 + 24 = 48 mm, then the top of hole II would be 62 + 48 = 110 mm above the base of piece IV, which indicates that the top of hole II is higher than the top of piece I, which does not appear so from this drawing.

Maybe this object was designed by M.C. Escher, or at least drawn by him for this problem.

Last edited by a moderator: May 7, 2017
18. Nov 6, 2015

haruspex

Good point. This seems to prove that the diagram is badly drawn that the student cannot be expected to get the answer.

19. Nov 6, 2015

SteamKing

Staff Emeritus
I agree that the diagram is badly drawn.

I don't agree that the student cannot be expected to get the answer. That's just excusing the student from thinking about what would be a reasonable interpretation of this figure, warts and all.

20. Nov 8, 2015

goldfish9776

if the radius of II is 24 , then the y bar of II should be 50+12+24 = 86 , am i right ?