Figure shows a cylindrical tube of radius `5cm` and length `20cm`. It is closed by a tight-fitting cork. The friction coefficient between the cork and the tube is `0.20`. The tube contains an ideal gas at a pressure of `1atm` and a temperature of `300K`. The tube is slowly heated and it is found that the cork pops out when the temperature reaches `600K`. Let `dN` denote the magnitude of the normal contact force exerted by a small length `dl` of the cork along the periphery (see the figure). Assuming that the temperature of the gas is uniform at any instant, calculate `(dN)/(dl)`.

Explanation

`(P_(1)V_(1))/(T_(1)) = (P_(2)V_(2))/(T_(2))`
`(1 xx V)/(300) = (P_(2) xx V)/(600)`

`P_(2) = 2 atm = 2 xx 10^(5) N//m^(2)`
Extra pressure `= P_(2) - P_(1) = 1 xx 10^(5) N//m^(2)`
`(mu N)/(A) = P_(2) - P_(1)`
`(0.2 xx N)/(pi r^(2)) = 1 xx 10^(5) rArr N = 10^(6) pi r^(2)`
`(d N)/(d l) = (N)/(2 pi r) = 5 xx (10^(5) pi r^(2))/(2 pi r)`
`= (5 xx 10^(5) xx 5 xx 10^(2))/(2)`
`= 1.25 xx 10^(4) N//m`