Question

A tall vertical cylinder of height $h$ and cross-sectional area $A$ is in uniform gravitational field. The cylinder contains an ideal gas at uniform temperature and is closed at both ends. If the density of the gas at bottom and top of the cylinder is ${\rho }_1$ and ${\rho }_2$ respectively, calculate the mass of the gas contained in the cylinder.

Answer 1

The distance $m$ density is given as

$S=S_1+\left(\frac{S_2-S_1}{n}\right)x$

$mass=S\times V$

$\int dm=Sa/V$

$M=(S_1+\left(\frac{S_2-S_1}{h}\right)x)dx$

$={[S_{1}x+\left(\frac{S_2-S_1}{2h}\right)x^2]}_0^4$

$=S_{1}h+\frac{(S_2-S_1)h}{2}$

$=(S_1+S_2)\frac{h}{2}$.