A practical application for cube root



#2

HP Forum readers --

Many low-end calculators have a built-in cube root function, which was added to the HP-33S as well. Consensus is that practical applications for the cube root function are limited.

There's certainly one use for it -- determining the edge of a cube having a given volume. Aside from that, what else might there be?

Here's one -- the height of a geostationary satellite, which stays over the same equatorial spot on the earth's surface as it orbits the planet in the direction of the earth's rotation. How high is the orbit? Too low, and the satellite must have a higher angular speed to apparently "travel eastbound"; too high, and it slips behind to apparently "travel westbound". (Away from the equator, and the component of gravity not needed for centripetal acceleration pulls the satellite toward the equator.)

Assume a perfectly-spherical Earth that stays at a fixed point, with a satellite in a circular orbit at altitude "h", in whose plane the Equator lies.

R  (radius of Earth)         6.37 x 106 meter
w (angular speed of Earth) 7.2921 x 10-5 rad/s
M (mass of earth) 5.97 x 1024 kilogram
G (Gravitational Constant) 6.67259 x 10-11 m3*kg-1*s-2

h (height of satellite)

m (mass of satellite) -- irrelevant to the answer

The attractive force "FG" between the two objects:

G * M * m
FG = ---------
(R + h)2

This equals the required centripetal force "FC":

m * [w * (R + h)]2
FC = -----------------
(R + h)

Equating,

G * M * m
--------- = m * w2 * (R + h)
(R + h)2

Canceling "m" and combining,

G * M = w2 * (R + h)3

Finally isolating for "h",

h = cbrt ( G * M / w2) - R

= 3.578 x 107 meter

= 22,236 miles!

NOTE: "w" is 2*pi radians divided by the sidereal day (23 hours, 56 minutes, 4.091 seconds) converted to seconds.

More at the following link:

http://en.wikipedia.org/wiki/Geostationary

-- KS


Edited: 27 Jan 2007, 5:43 p.m. after one or more responses were posted


#3

Hello!

Quote:
Here's one -- the height of a geostationary satellite ...

Yes, but ... this height is the same for _all_ satellites of a given planet. And since the number of planets in reach of our spacecraft seems to be somewhat limited, this application does hardly justify a cube-root key on every calculator ;-)

Greetings, Max


#4

Hi, Max --

Quote:
Yes, but ... this height is the same for _all_ satellites of a given planet. And since the number of planets in reach of our spacecraft seems to be somewhat limited, this application does hardly justify a cube-root key on every calculator ;-)

Of course -- I agree completely, but it's one of the very few specific applications I've encountered that require a cube-root solution. I worked through it in the early 1980's, learning about physics and meteorological satellites (which also are geostationary). Knowing how distant they are, I'm very impressed that these satellites have been utilized so effectively -- even though a relayed signal will take just under a quarter-second round-trip.

Another third-power equation involves the power required for a land vehicle to overcome aerodynamic losses in calm air (making air speed = ground speed).

-- KS

#5

It always amused me why some machines use precious keyboard space for such a function, and also for the "xth root of Y" function. Note that vintage HP's required the user to muster enough mathematical wherewithall to use the 1/X, Y^x key combination, which was dead easy, as these two keys were in very close proximity (if not next to each other) on most early HP machines.
I suppose one could argue for the abolishment of the square root key by this same logic, but let's not go too far...:)

Best regards, Hal


#6

Could the cube root function be a hangover from slide rule days when a cube root scale had more applications than the cube root function on a calculator?

On a slide rule the cube root scale was more often used to find cubes with the answer lying on the main D scale where it was convieniently multiplied by other factors. For example to find the volume of s sphere you would set the cursor on the radius or diameter on the cube root scale and then multiply by the appropriate factors on the C & D scales.

The average slide rule did not have a cube root scale, they were seen on "muscle" slide rules such as the Pickett N3-ES.

As for using the cube root function on a calculator, there are certain effects in the near field of an antenna that employ a cube root. One might wish to find the radius of a given sphere, etc. I seem to remeber having to use cube root somewhere along the line in calculus.


#7

a lot of early machines performed X^Y using logs without special treatment. so (-27)^(1/3) often failed where cbrt(-27) worked.

also, i've seen machines with a more accurate cube rooter than the same machine gives with logs, indicating a special implementation.

#8

The Manning formula for open channel flow requires the hydraulic radius to be raised to the (2/3) power. I have a Hemmi 269 civil engineering rule with a K (cubic) scale for that purpose. It also has an F (fourth power) scale, but I don't remember why.


#9

I'm looking at my 1962 Pickett slide rule Model N 500 ES right now and see a "K" scale. I don't think I ever used it.

tm


#10

PS

Then I thought about the 6 inch (Model N600-ES). Wow, it also has a "K" scale!

tm


#11

Pickett "6 inch (Model N600-ES). "

Best slide rule ever (or... the only one I currently take with me.)

Edited: 25 Jan 2007, 11:18 p.m.


#12

Quote:

Best slide rule ever (or... the only one I currently take with me.)


If you had to do cubes and cube roots why not use a P&E Model 2?

#13

Quote:
The Manning formula for open channel flow requires the hydraulic radius to be raised to the (2/3) power. I have a Hemmi 269 civil engineering rule with a K (cubic) scale for that purpose. It also has an F (fourth power) scale, but I don't remember why.
In the (unlikely) event that anyone cares why a civil engineering rule has x^3 (K) and x^4 (F) scales, in addition to the standard x^2 (A/B) and x^1 (C/D) scales, here are the answers.

(1) Manning's formula, which is used to determine the velocity of flow in pipes and channels, uses an R^(2/3) term, where R is the hydraulic radius. The K and A scales can be used together to calculate R^(2/3), with the answer on the A/B scales.

But you also have to multiply R^(2/3) by S^(1/2), where S is the gradient. Normally square roots would be calculated by entering a value on the A/B scales, and reading the square root on the C/D scales. But in this case, it is more convenient to enter S on the F scale, so that the square root of S falls on the A/B scales. That way both R^(2/3) and S^(1/2) use the A/B scales, and can be readily multiplied.

(2) Both F and K scales are convenient for calculating moments of inertia (e.g. for beams or columns). The formulas for moments of interia commonly involve x^3 or x^4 terms

Edited: 27 Jan 2007, 1:23 p.m.

#14

Quote:
It always amused me why some machines use precious keyboard space for such a function, and also for the "xth root of Y" function.

I am glad XROOT is available on some calculators. On the HP-42S

27 +/- ENTER 3 1/x yx
returns 1.5 i2.59807621135. This is not wrong but it is not the answer I usually want (The other answers are 1.5 -i2.59807621135 and -3). Likewise on the HP-50G
-27 3 1/x yx
returns (1.5, 2.59807621135) but
-27 3 XROOT 
will return -3, which might please most users.

Best regards,

Gerson.

#15

Hi, Hal --

As Gerson suggested, the "y^(1/x)" function allows the real-valued odd root of a negative real argument to be found directly. Though there is a simple workaround, y^(1/x) may be handy in a program for that reason.

y^(1/x) can also be slightly more accurate than [1/x][y^x] for certain arguments, due to internal extended precision.

However, x3 and x1/3 really don't add much value...

-- KS

#16

Karl, you made a mistake in the dimension of G

Quote:
G (Gravitational Constant) 6.67259 x 10-11 m3*kg-2*s-2

It is just kg in the denominator.

#17

I note that my HP-48GX returns (1,1.732...) for -8^1/3 which is the principle root. For 3V-8 it returns -2 which is the answer most people expect but it is the second root and not really correct since the surd is supposed to return the principle root and only the principle root.

Sloppiness in signs and which root you are playing with can lead to trouble. I have a "proof" that -1 = 1. the fallacy is extremely subtle. I have seen published explanations of the fallacy which themselves were fallicious. In high school I drove my student math teacher to tears with it.


#18

Quote:
I note that my HP-48GX returns (1,1.732...) for -8^1/3 which is the principle root. For 3V-8 it returns -2 which is the answer most people expect but it is the second root and not really correct since the surd is supposed to return the principle root and only the principle root.

I had to look up "surd": An archaic term for an irrational number.

I'd say that this approach, although mathematically sound, is rather restrictive. If the physical problem being addressed resides entirely in the domain of real numbers, then the real-valued root is always of primary interest, even though -- if negative -- it is not the principal root.

Only a "y^(1/x)" function ("XROOT" on the HP-48/49) can directly provide the negative-valued odd root of a negative real number.

While the HP-32SII has "y^(1/x)", the cube-root workaround for an HP-32S (or HP-15C, HP-11C, HP-34C, etc.) program might be as follows:

CF 0
x<0?
SF 0
FS? 0
+/- (or ABS)
3
1/x
yx
FS? 0
+/-
CF 0

This could be approached similarly on an HP-41 or HP-42S, or the user could create two separate branches for y<0 and y>=0 with local labels.

-- KS


Edited: 28 Jan 2007, 3:50 p.m.


#19

"surd" is also an older term for what we call today the radical sign or sometimes square root sign. I think there is an entymological (did I spel lthat right?) relationship by way of the fact the square root of 2 is a famous irrational.


As a function, the surd or radical is supposed to return the principal and only the principal root, it's a single valued function. That's why we have to put a plus minus sign in front of it when we mean both square roots. I suppose that the HP-48 returns -2 for 3V(-8)since we are so often interested in the real valued root. (V for the radical or surd, hard to replicate in plain text.) But it's also a pit fall that will lead to errors in some situattions. Once again proves you ought to know what you are doing and not just engage in mindless number crunching.


#20

etymological

#21

Hi, Karl:

Karl posted:

    "I had to look up "surd": An archaic term for an irrational number."

      It appears all the time in Srinivasa Ramanujan's papers, as well as other mathematicians of the time such as Hardy, Klein, etc., etc.

Best regards from V.
#22

Hi, Marcus --

Yes, you are correct -- thank you. I got a bit sloppy with the algebra, there...

The equation is corrected.

It's quite understandable that this topic attracted the notice of "Mr. von Cube" :-)

-- KS

Edited: 27 Jan 2007, 10:36 p.m.

#23

to calculate the height of snow drift on a roof you use the cube root and also take another term to the 1/4 power. This is essential information when designing the framing for a roof


#24

ny k & e Deci-log, 1967 version) has a k scale., but at my university the rule was mostly used by econmics and liberal arts because of the extra log scales facilitating compound interest problems and population growth.


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