## TuringTest 3 years ago 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?

1. TuringTest

|dw:1347649715746:dw|

2. TuringTest

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

3. amistre64

what are the steps you are taking?

4. mukushla

what is magnitude of d comparing with R ?

5. TuringTest

|dw:1347649869832:dw|$V=\frac43\pi(R^3-r^3)\approx4(R^3-r^3)$$d=R-r$

6. amistre64

are you integrating the surface area from r to R ?

7. TuringTest

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

8. TuringTest

why am I integrating?

9. akash809

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

10. TuringTest

I tried and it's not seeming obvious

11. TuringTest

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$

12. TuringTest

R^2d I mean...

13. TuringTest

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

14. TuringTest

oh I should have tried difference of cubes formula maybe?

15. mukushla

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$V=\frac{4}{3}\pi(R^3-r^3)=\frac{4}{3}\pi d(R^2+R.R+R^2)=4\pi d R^2$

16. TuringTest

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!

17. mukushla

np

18. phi

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

19. TuringTest

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

20. phi

oh... see mukushia

21. TuringTest

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

22. phi

yep, a typo