Re: A newcomer here

On Mon, 12 Sep 2005 23:03:07 -0400, "Brian M. Scott"
<b.scott@xxxxxxxxxxx> wrote:

>On Tue, 13 Sep 2005 02:03:27 GMT, Logan Kearsley
><chrono.surfer@xxxxxxxxxxx> wrote in
><PrqVe.6937$b37.3139@trnddc04">news:PrqVe.6937$b37.3139@trnddc04> in
>rec.arts.sf.composition:
>
>> "David Friedman" <ddfr@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message
>> news:ddfr-DAA931.18240812092005@xxxxxxxxxxxxxxxxxxxxxxxx
>
>>> In article <GepVe.3508$XO6.2586@trnddc03>,
>>> "Logan Kearsley" <chrono.surfer@xxxxxxxxxxx> wrote:
>
>>>> With constant crust thickness and density, the surface gravity
>>>> increases as the radius increases.
>
>>> No. It stays constant.
>
>>> Area goes as the square of radius, so with constant thickness and
>>> density mass goes as the square of radius.
>
>> Except that it doesn't, exactly. It's somewhat off. Assuming constant
(snip)

>Let inner radius = r, outer radius = r + t. Then volume =
>(4/3)*pi*[(r + t)^3 - r^3] = (4/3)*pi*(3tr^2 + 3rt^2 + t^3),
>i.e., mass is proportional to 3tr^2 + 3rt^2 + t^3, and force
>is proportional to 3t + 3t^2 / r + t^3 / r^2, which for
>constant t decreases as r increases.
>
>Alternatively, let the inner radius be r, the outer radius
>kr. The volume is proportional to (k^3 - 1)r^3, the force
>to (k^3 - 1)r = (1 + k + k^2)(k - 1)r = (1 + k + k^2)t,
>since t = (k - 1)r. Thus, the force is proportional to
>1 + k + k^2. But if t is fixed, k = 1 + t/r is decreasing
>as r increases, so the force is decreasing.

Force goes as the inverse square of the *outer* radius -
it's a small point, but I think it changes the sign of
the t^2 term ... e.g. if you put [t^3 - (r-t)^3] in place
of your original [] factor.

The more interesting question is: how does the thickness
have to vary for a constant force? From the formulae
above, it looks like the thickness has to increase, but
if you change the sign of the second term in each case,
it has to decrease, unless t is comparable to r, which
it was starting with a solid sphere :-).

I think I have to resort to a scratchpad ... I can't
solve cubic equations in my head this morning :-)

Jonathan
(Or any other morning, come to that!)

--
Mail to spam auto-deleted, use jlc1 instead.
(That's jay ell cee one, if your font makes l and 1 look the same)

.

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