- #1

Elias Waranoi

- 45

- 2

## Homework Statement

A vertical cylinder of radius r contains an ideal gas and is fitted with a piston of mass m that is free to move. The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is p

_{0}. In equilibrium, the piston sits at a height h above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance h + y above the bottom of the cylinder, where y << h.

## Homework Equations

F = pA = ma

pV = nRT (pressure p, volume V, moles in gas n, gas constant R, temperature T)

V = πr

^{2}h

## The Attempt at a Solution

I managed to solve a) by myself and get p

_{1}= p

_{0}+ mg/πr

^{2}

Trying to solve b)

Positive force is upwards so ∑F = F - F

_{0}- F

_{p}where F is the force upwards on the piston by the pressure of the gas underneath, F

_{0}is the downward force on the piston by the air pressure p

_{0}above and F

_{p}is the downward force by the weight of the piston.

F

_{0}= p

_{0}A = p

_{0}πr

^{2}

F

_{p}= mg

moles in ideal gas n = p

_{1}V

_{1}/RT

_{1}(1)

pressure p

_{2}= nRT

_{2}/V

_{2}(2)

Because of heat conduction and the constant temperature outside the cylinder, T

_{2}= T

_{1}. Inserting (1) in (2) yields:

p

_{2}= p

_{1}V

_{1}/V

_{2}

V

_{1}= πr

^{2}h

V

_{2}= πr

^{2}(h + y)

p

_{2}= p

_{1}h/(h + y)

∑F = p

_{2}A - p

_{0}A - mg. A = πr

^{2}.

Simplifying yields my answer: ∑F = (h/(h + y) - 1)(mg + p

_{0}πr

^{2})

Textbook answer: ∑F = -(y/h)(mg + p

_{0}πr

^{2})

I tried replacing the variables with values to see the result of both answers. My answer is not correct apparently, what did I do wrong?