- #1
K29
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I have a spherical shell with inner radius [itex]a_{1}[/itex] and outer radius [itex]a_{2}[/itex]
I have worked out that the tension at [itex]r=a_{1}[/itex] is
[itex]\tau=\frac{2a_{1}^3+a_{2}^3}{2(a_{2}^{3}-a_{1}^{3})}P_{1}[/itex]
([itex]P_{1}[/itex] is pressure from the inside of the shell, causing the tension.
Now if the shell is not very thick. [itex]t=a_{2}-a_{1}[/itex] is small. [itex]\frac{t}{a_1}<<1[/itex]
and I should be able to show
[itex]\tau\approx\frac{a_{1}}{2t}P_{1}[/itex]
But I am not sure about the first step to take in getting there. Any ideas? Please help.
I have worked out that the tension at [itex]r=a_{1}[/itex] is
[itex]\tau=\frac{2a_{1}^3+a_{2}^3}{2(a_{2}^{3}-a_{1}^{3})}P_{1}[/itex]
([itex]P_{1}[/itex] is pressure from the inside of the shell, causing the tension.
Now if the shell is not very thick. [itex]t=a_{2}-a_{1}[/itex] is small. [itex]\frac{t}{a_1}<<1[/itex]
and I should be able to show
[itex]\tau\approx\frac{a_{1}}{2t}P_{1}[/itex]
But I am not sure about the first step to take in getting there. Any ideas? Please help.