mean radius of a cylinder?

Question

Please tell me how to calculate the mean radius of a cylinder whose radius and height are known.

an example you can use to try this yourself is radius=10 height=20 and the mean radius is 11.32.

I am asking for the MEAN RADIUS.

this requires an integral.

I am asking for the average radius over all points from the midpoint of a cylinder.

don't you fools understand this problem requires CALCULUS? Are there any people on here that understand complicated mathematics and can think critically?

Answer 1

Mean radius is the average distance from the center of the object to the surface. For a sphere, it is, of course, a constant.

You are correct that it usually requires calculus (and I don't have a formula for you off the top of my head), but sometimes there is a trick that simplifies it into a simple expression with no calculus. Consider the symmetry of the cylinder, and you just might come up with one.

Given the circular symmetry about the axis, I would start with an arbitrary slice (or "half=slice") thru the center. Just imagine the surface as a line of length 20, with the center a distance of 11.32 away from its midpoint. Now, what is the average distance?

I can imagine two identical right triangles (one going up, one going down), so all we really need to do is integrate within one triangle to find the average distance from the vertex (representing the center of the cylinder), along the line from 0 to 10, as it goes from 11.32 to whatever is the hypotenuse of the right triangle (with sides of 11.32 and 10. -- that should be pretty easy, and I'll leave it to you!)

Better yet, since there are as many values above the half-way point as there are below, it seems to me the mean is simply the hypotenuse of a right triangle with sides of 11.32 and 5.

(Geez, I wish it was 12 and 5, since that would make the answer 13.)

Just solve for sqrt( 11.32^2 + 5^2)
=sqrt( 153.14 )
= 12. something
[Sorry, I don't have a calculator handy ]

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I hope that reduces the problem to one you can solve
(without calculus, since you are integrating along a straight line, to find the mean of a linear function -- which should be the same as the midpoint!)
.






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Answer 2

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I answered this question in detail before. See below. I can only reiterate that your weighting of the mean is unusual at best. Here is the previous answer: The use of the term "mean" here is pretty odd. You are calculating the radius to points on the surface of a cylinder but you are not weighting the values based on the surface but instead based on the differential area of a spherical surface. I only discovered this by implementing weird interpretations of the term "mean" until I got the coveted "right" answer. I see no physical reason to use a spherical surface in your weighting although for some specific problem it could be valid. In that case you should state it instead of using the nebulous "mean". Otherwise, you could choose a cube or an ellipsoid for your integration surface and get different but equally valid answers. That aside, here is how you get your answer: You need to use a spherical surface differential element (r^2 * cos(phi)* dphi * dtheta with phi as the latitude, not colatitude, and theta as the azimuth) and a definition of the radius in spherical coordinates for the walls (r/cos(phi)) and lid (h/(2 * sin(phi))) separately. The resulting integrals are pretty simple. The two integrations are simply added to produce the weighted sphere area-cylinder radius product. Since the whole thing is weighted on the spherical area, you need to divide by the sphere area to get the mean. The closed form relation for any such spherically-weighted-mean cylinder radius is: Define an aspect ratio: k = h/(2*R) Mean Radius = R*(atan(k) + (k/2) *Ln(1 + 1/(k^2)))

Answer 3

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RE:
mean radius of a cylinder?
Please tell me how to calculate the mean radius of a cylinder whose radius and height are known.

an example you can use to try this yourself is radius=10 height=20 and the mean radius is 11.32.

Answer 4

The radius of a cylinder is constant. So if the radius = 10, then the mean radius is 10.

Answer 5

this is tricky question...is radius of oc cylinder is constant...so mean radius of the cylnder is the same as the radius of the cylinder so if the radius is 10 then the mean radius is 10 also...
anyhow try to differntiate betwwen the two: mean raduis & radius of cylinder...i hope it help this information....