What is the Correct Calculation for Centroid of Composite Area?

[goldfish9776]

Homework Statement

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 ?

Homework Equations

The Attempt at a Solution

 

[haruspex]

1. the y bar for I should be 16.98, am i right ?

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.

 

[goldfish9776]

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.

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

 

[haruspex]

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

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?

 

[goldfish9776]

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?

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 ?

 

[SteamKing]

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?

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 ?

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.

 

[SteamKing]

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?

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.

 

[haruspex]

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.

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.

 

[SteamKing]

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.

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.

 

[haruspex]

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.

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.

 

[SteamKing]

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.

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.

 

[haruspex]

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.

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.

 

[SteamKing]

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.

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

 

[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 isn't at the center of the circle , but rather on the tangent of the circle ?

 

[SteamKing]

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 isn't at the center of the circle , but rather on the tangent of the circle ?

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.

 

[haruspex]

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.

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.

 

[SteamKing]

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.
View attachment 91454

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.

http://www.allartnews.com/wp-content/uploads/2011/02/M.C.-Escher-Relativity-1953-Lithograph.jpg​

 

[haruspex]

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.

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

 

[SteamKing]

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

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.

 

[goldfish9776]

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.

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

 

[haruspex]

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

SteamKing has proved that if you take the 50mm as being the distance from the top of the base plate to the bottom of the cut-out circle (II) then the top of that cut-out circle would be above the backplate, clearly not possible. So you need to interpret the 50mm as the height of the centre of II above the baseplate: 50+12=62. It's just very badly drawn.