Question

Why can I not seem to show that the volume of a spherical shell is\[V\approx4R^2d\]where R is the outer radius and d is the thickness?

Answer 1

This can't be that hard, I must be doing something stupid.

Answer 2

what are the steps you are taking?

Answer 3

what is magnitude of d comparing with R ?

Answer 4

\[V=\frac43\pi(R^3-r^3)\approx4(R^3-r^3)\]\[d=R-r\]

Answer 5

are you integrating the surface area from r to R ?

Answer 6

well it's not supposed to be negligible. this is supposed to be a rough model of the earth were it covered completely in water, but I don't think I am to assume that d is that small that we can drop it from the calculations

Answer 7

why am I integrating?

Answer 8

looks like u are using the right formula...expand R^3-r^3 ..using formula of a^3-b^3..see if u get something on expanding

Answer 9

I tried and it's not seeming obvious

Answer 10

ignoring the 4 out front I want to get the r^2d part...\[r=R-d\]\[R^3-(R-d)^3=3R^2-3Rd^2+d^3\]

Answer 11

R^2d I mean...

Answer 12

\[V\approx4(3R^2-3Rd^2+d^3)\approx4 R^2d~~~???\]there must be a better way to see this...

Answer 13

oh I should have tried difference of cubes formula maybe?

Answer 14

emm..\[V=\frac{4}{3}\pi(R^3-r^3)=\frac{4}{3}\pi(R-r)(R^2+rR+r^2)\]thats it max

let suppose d<<R so \(r = R\) so\[V=\frac{4}{3}\pi(R^3-r^3)=\frac{4}{3}\pi d(R^2+R.R+R^2)=4\pi d R^2\]

Answer 15

This is why I'm practicing this kind of reasoning, I would not have thought to use \(r=R\) at that point. The MIT problems are tough; estimating the mass of water on the earth...
Thanks mukushla!

Answer 16

np

Answer 17

I get 4 pi d R^2 not 4 d R^2
for R ~ r

Answer 18

that's what I was about to say... maybe there is a typo on the doc; there is a weird space after the 4 (check 2.3.5)
http://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/problem-solving-and-estimation/MIT8_01SC_coursenotes02.pdf

Answer 19

oh... see mukushia

Answer 20

ah yes, the next line has the approximation as 4piR^2d

Answer 21

yep, a typo