A fixed thermally conducting cylinder has a radius `R` and height `L_(0)`. The cylinder is open at its bottom and has a smaller hole at its top. A piston of mass `M` is held at a distance `L` from the top surface, as shown in the figure. The atmospheric pressure is `P_(0)`. While the piston is at a distance `2L` from the top, the hole at the top is sealed. The piston is then released to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is
A
`rhog(L_0-H)^2+p_0(L_0-H)+L_0p_0=0`
B
`rhog(L_0-H)^2-p_0(L_0-H)-L_0p_0=0`
C
`rhog(L_0-H)^2+p_0(L_0-H)-L_0p_0=0`
D
`rhog(L_0-H)^2-p_0(L_0-H)+L_0p_0=0`
Text Solution
Verified by Experts
The correct Answer is:
C
`p_(1)=p_(2)` `p_(0)+pg(L_(0)-H)=p` (i) Now, applying `p_(1)V_(1)=p_(2)V_(2)` for the air inside the cylinder, we have `p_(0)(L_(0))=p(L_(0)-H)` `p=(p_(0)L_(0))/(L_(0)-H)` Substituting in Eq. (i), we have `p_(0)+pg(L_(0)-H)=(p_(0)L_(0))/(L_(0)-H)` `implies pg(L_(0)-H)^(2)+P_(0)(L_(0)-H)-L_(0)p_(0)=0` Therefore, option `(c)` is correct.
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