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A non conducting piston of mass m and ar...

A non conducting piston of mass m and area of cross section A is placed on a non conducting cylinder as shown in figure. Temperature, spring constant, height of the piston are given by T,K,h respectively. Initially spring is relaxed and piston is at rest. Find
(a) Number of moles
(b) Work done by gas to displaced the piston by distance d when the gas is heated slowly.
(c ) Find the final temperature

Text Solution

Verified by Experts

a. `PV = nRT implies (P_(atm) + (mg)/(A)) Ah = nRT`
`implies n = ((P_("atm") + (mg)/(A))Ah)/(RT)`
b. 1st method : Applying newton's law on the piston
`mg + P_("atm")A + Kx = P_("gas")A`
`W_("gas") = int_(0)^(d) P_(g)A dx = int_(0)^(d) (mg + P_("atm")A + Kx) dx`
`implies W_("gas") = mgd + P_("atm")dA + (1)/(2) Kd^(2)`
2nd method : Applying work energy theorem on the pistion
`W_("all") = Delta KE`
c. Since pistons moves slowly therefore `Delta KE = 0`
`W_("gravity") + W_("gas") + W_("atm") + W_("spring") = 0`
`- mgd + W_("gas") + (- P_("atm")Ad) + [-((1)/(2) Kd^(2) - 0)] = 0`
`implies W_("gas") = mgd + P_("atm")dA + (1)/(2) Kd^(2)`
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