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Re: [Phys-L] [ext] rms / conic / arithmetic / geometric averages

I've been enjoying this thread, which reminds me a little of an animated demonstration my brother and I created last year:

Take three hollow, open-top objects: a cone, a hemisphere, and a cylinder, all with radius = height. (Yes, that would be obvious for the hemisphere.) This radius and height are the same for all three containers, so the open top surface areas are the same. Fill them with water.

The volumes are related as 1:2:3.

If you add water at the same constant rate, 6 full cones = 3 full hemispheres = 2 full cylinders.

Of course, when the containers are filling, the fluid height grows at different rates: V/V_full_cone = 3 f (for cylinder), = f^3 (for cone), = f^2 (3-f) (for hemisphere), where f = h/r.


Kenneth E. Caviness, Ph.D.
Chair & Professor, Physics & Engineering
Southern Adventist University 
P.O. Box 370, Collegedale, TN 37315
Office:   423-236-2856
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-----Original Message-----
From: Phys-l <> On Behalf Of John Denker via Phys-l
Sent: Saturday, 22 May, 2021 17:18
To: Forum for Physics Educators <>
Cc: John Denker <>
Subject: [ext] [Phys-L] rms / conic / arithmetic / geometric averages

Hi --

I recently noticed something cute about averages. Maybe everybody but me already knew this, but I was surprised.

Context: Suppose you have a round pot or bucket with uniformly tapering sides. In other words, it is a truncated cone. You know the height, and you know the ID at the bottom and at the top.

We all know the formula for the volume of a cylinder, and this is "almost" a cylinder, so maybe we can use the same formula, using some sort of *average* radius.

What sort of average should we use? RMS? Arithmetic? Geometric????

For a full cone, we know the volume is 1/3 of the volume of a cylinder with the same diameter and radius. So RMS is no good, because that would predict 1/2 of the volume rather than 1/3.
And arithmetic is no good, because it would predict 1/4 of the volume (half of the diameter) instead of 1/3 of the volume.
Geometric is even worse, because it would predict 0 instead of 1/3. We want something that is somewhere between RMS and arithmetic.

The cute thing is that all of the above form a family. You can interpolate between them. The general formula is:
average = √[(Bot2+N*Bot*Top+Top2) / (2+N)]

for all N from 0 to ∞. In particular:

N=0 --> RMS
N=1 --> conic
N=2 --> arithmetic
N=∞ --> geometric

The relationships are graphed here:

The N=1 average is in fact exactly correct for a truncated cone, as you can easily verify in lots of ways. (Grind out the integrals.
Or just subtract one full cone from another.)

Related: Here is an online calculator to figure out the volume of a truncated cone:
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