Calling all mathematicians!!

Fergie

Fergie

Fergie

Associate
Joined
11 Jan 2005
Posts
515
Good afternoon to you all.

I know a few of you are maths bods so I thought I'd bring my problems to you :p



This is what I have, When half-sphere is in contact with the cone, it blocks the flow of liquid coming from the pipe that it is connected to. I want to the resultant area of the sphere that the liquid will be acting on.



Here are the dimensions. I've been trying to use catia to solve it for me, but my skills leave a lot to be desired. :(

I've been searching the net for formulae and methods, but they all go way over my head or don't do the job.

Cheers for the help. :)
 
Surface area of part of a sphere = 2piR x the "length of part sphere"; where the "length of part sphere" is basically the length of the bit of sphere you want the surface area for, measured along the 'x' axis in your diagram. So, for example, the whole sphere would have "length of part sphere" = 2R, hemisphere would be R, etc).

If I'm understanding your diagram, correctly, the "length" here is R sin 60 = R sqrt(3)/2.

So the final answer would be pi x R^2 x sqrt(3).

Not totally sure that is the the correct value for the length though, so it would make sense for you to check it if you can easily.
 
Excuse the crappy picture, anyway from what I understand the head of the round bit can be equated to a sphere so you can replace it with something like this...

pipems8.jpg


Now if you work out the angle between the points of contact between the sphere and the inside of the valva (A and B on the diagram) then you should be able to use the formula

Area = (X / 360) * (pi * r)^2

Where X is the angle between A and B and r is the radus of the sphere
 
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Right, well I've got A=2πr²(1-sin(θ/2)), where r is the radius, and θ is the angle (60º in this case).

That gives a surface area of 14.2 square units with your measurements.

Basically you need to find the distance h between the furthest-in edge of the sphere in the socket and the plane that cuts across between where the sphere's surface makes contact with the socket surface. You can then use the formula A=2πrh to find the area of this cap that's inside the socket.
 
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Spuds said:
Area = (X / 360) * (pi * r)^2

Where X is the angle between A and B and r is the radus of the sphere
Click to expand...
That's the formula for the area of a sector of a circle, not the surface area of part of a sphere.
 
Inquisitor said:
Right, well I've got A=2πr²(1-sin(θ/2)), where r is the radius, and θ is the angle (60º in this case).

That gives a surface area of 14.2 square units with your measurements.
Click to expand...

I agree with the result but got it differently.

Taken from a random page about spheres: (http://www.mathpages.com/home/kmath343.htm)

343fig3.gif


"Archimedes' theorem is that the surface area of the region of the
sphere below the horizontal plane H is equal to the area of a circle
of radius t."

In your case you can easily demonstrate that 't' is actually equal to the radius of your sphere (A handy side effect of your 60 degree cone) meaning that the area of the section of sphere is pi*r^2 = 14.2

I quite like that problem. Seems rather complicated on the outside but actually has an incredibly elegant, simple solution.
 
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DaveF said:
That's the formula for the area of a sector of a circle, not the surface area of part of a sphere.
Click to expand...

Ah... My bad. Try Area = pi * (h^2 * r^2)

Where h is the height of the dome (i.e. The tip of the sphere to the points of contact) and r is the radius of tghe sphere
 
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