Question

A thin heavy metal plate is being bombarded by a perpendicular beam of gas particles from both sides as shown in the figure. The solid dots are representing the molecules hitting from left side and the faint dots are the molecules hitting from right side. The mass of these gas particles is $m={10}^{-26}kg$ and velocity before hitting is $v_0=5m/s$. Volume density of the gas particles on both sides is $n={10}^{25}$ per $m^3$. Each beam has an area $A=1m^2$ and the collisions are perfectly elastic. What is the external force $F$ (in newton) required to move the plate with a constant velocity $v=2m/s$

Answer 1

$f=$ change in momentum per second $=\frac{dp}{dt}$

$=\frac{\{2dm\left(v_1\right)-2dm\left(v_2\right)\}}{dt}$

$\Rightarrow dm=m\eta dx$

$\Rightarrow v_1=$ relative velocity of one side of gas

$v_2=$ relative velocity of other side gas

$\Rightarrow f=2m\eta A\frac{dx}{dt}{v}_1-2dmA{v}_2\frac{dx}{dt}{x}^1$

$=2m\eta A{v_1}^2-2m\eta A{v_2}^2$

$\Rightarrow f=2m\eta A({v_1}^2-{v_2}^2)$

$=2m\eta A\{{(5+2)}^2-{(5-2)}^2\}$

$=2\times {10}^{-26}\times {10}^{25}\times 1(49-4)$

$=0.2\times 40$

$=-8N.$

Hence, the answer is $-8N.$