2.6.53 | before revision #13300 | Savchenko Solutions

Statement

2.6.53^∗. The famous physicist F. Dyson suggested that it would be possible to fully utilize the energy of stars if space civilizations could surround stars with spherical shells. Find the stress in the material of a stationary homogeneous shell that would surround the Sun according to this assumption, with its radius equal to the radius of the Earth’s orbit. The density of the shell material $$ρ =4\cdot10^3 kg/m^3$$ .

Solution

$$$σdS=dF$$$

$² $$S=R²Ω$$$

$$$dS=2RdRΩ$$$

$dR=\frac{dS}{2RΩ}$

$$$dF=g*dm$$$

$\oint gdS=-4πGM$

$² $$-4πR²g=-4πGM$$$

$²$$g=\frac{GM}{R²}$$$

$$$dm=ρdV$$$

$²$$dV=4πR²dR=\frac{RdS}{2}$$$

$²$$σdS=\frac{GM}{R²}ρ\frac{RdS}{2}$$$

$σ=\frac{GMρ}{2R}$

$¹²$$\boxed{σ=1.8*10¹²Pa}$$$

Answer

$¹ ² $$σ=1.8*10¹²Pa$$$