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Solved: Oscillations of a Piston. A vertical cylinder of
Chapter 19, Problem 69CP(choose chapter or problem)
CP Oscillations of a Piston. A vertical cylinder of radius contains a quantity of ideal gas and is fitted with a piston with mass
that is free to move (Fig.
). 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
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
is much less than
. (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?
Equation Transcription:
Text Transcription:
p_0
h+y
p_0
Questions & Answers
QUESTION:
CP Oscillations of a Piston. A vertical cylinder of radius contains a quantity of ideal gas and is fitted with a piston with mass
that is free to move (Fig.
). 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
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
is much less than
. (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?
Equation Transcription:
Text Transcription:
p_0
h+y
p_0
ANSWER:
Solution 69CP
a.) The cylinder is placed in a constant temperature bath, so the process is isothermal.
At equilibrium,the difference between the pressure inside and outside the cylinder, multiplied by the area of the piston, must be the weight of the piston.
() A= m
= (m
/A
P = + (m
/A
Therefore, the absolute pressure of the gas trapped below the piston when in equilibrium is P = + (m
/A
b.)
Step 1 of 2:
When the piston is a distance (h+y) above the cylinder, the pressure in the trapped gas is
P =(h/h+y)
for values of y small compared to h
= (1+
)