Question

A long container has air enclosed inside at room temperature and atmoshperic pressure `(10^(5) Pa)`. It has a volume of 20000 c c. The area of cross section is `100 cm^(2)` and force constant of spring is `k_("spring") = 1000 N//m`. We push the right piston isothermally and slowly till it reaches the origianl position of the left piston which is movalbe. Final length of air colume is found to be 25 h cm. Assume that spring is initially relaxed. Find `h`.

Answer 1

Correct Answer - 4
`P_(2) V_(2) = P_(1) V_(1)`
`V_(1) = 20000 c c P_(1) = 10^(5) Pa`
`V_(2) = Ax, P_(2) = P_(0) + (kx)/(A)`
`implies 10^(5) xx 20000 xx 10^(-6)`
`= (10^(5) + (1000 x)/(100 xx 10^(-4))) 100 x xx 10^(-4)`
`implies 2000 = 1000 [x (1 + x)]`
`x^(2) + x - 2 = 0`
`x = (-1 +- sqrt(1 + 8))/(2) = 1 m = 100 cm`
`implies 25 h = 100 implies h = 4`