An ideal gas at `NTP` is enclosed in an adiabatic vertical cylinder havin an area of cross section `A=27 cm^(2)` between two light movable pistons as shown in figure. Spring with force constant `K=37-- N//m` is in a relaxed state initially. Now the lower position is moved upwards a distance `h//2`, `h` being the initial length of gas column . It is observed that the upper piston moves up by a distance `h//16`. Final temperature of gas` 4//3 xx 273 K`. Take `gamma` for the gas to be `3//2`.

The value of `h` is

Options

A. `1 m`
B. `1.4m`
C. `1.6m`
D. `2m`

Answer

C

Explanation

Final pressure,
`P_(f)=(P_(0)+(Kh//16)/(A))`
`(P_(f)V_(f))/( T_(f))=(P_(0)V_(0))/(T_(0))`
`((P_(0)+(kh)/(16A))Axx((h)/(2)+(h)/(16)))/(T_(f))=(P_(0)Ah)/( T_(0))`
`((10^(5)+(3700h)/( 16xx27xx10^(-4)))(9)/(16))/((4)/(3)xx273)=(10^(5))/(273)`
`(1+0.856h)=(16)/(9)xx(4)/( 3)`
Solving, `h=1.6 m `