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A frictionless piston of mass m is at a precise fit in the vertical cylinder neck of a large container of volume V. The container is filled with a gas and there is a vacuum above the piston. THe cross sectional area of the neck(and thereby the piston) is A.
a) Derive an equation for the pressure of the gas in the container when the piston is in equilibrium.
b) Assuming that the pressure and volume of the gas are related by boyle's law, derive an equation for the restoring force on the piston when it is displaced by a small amount of x.
c) assuming that the motion of the piston is small enough for boyle's law to be valid, obtain the differential equation for small displacements of the piston about its equilibrium position.
d) show that the angular frequency of oscillation, omega, is independent of m.
e) calculate omega for V=2000 liters and A= 1E-4 meters squared.
I am not to sure where to begin...I know boyle's law is PV = k where...
P denotes the pressure of the system.
V is the volume of the gas.
k is a constant value representative of the pressure and volume of the system
If you could help me in a direction to go that would be great thank you
a) Derive an equation for the pressure of the gas in the container when the piston is in equilibrium.
b) Assuming that the pressure and volume of the gas are related by boyle's law, derive an equation for the restoring force on the piston when it is displaced by a small amount of x.
c) assuming that the motion of the piston is small enough for boyle's law to be valid, obtain the differential equation for small displacements of the piston about its equilibrium position.
d) show that the angular frequency of oscillation, omega, is independent of m.
e) calculate omega for V=2000 liters and A= 1E-4 meters squared.
I am not to sure where to begin...I know boyle's law is PV = k where...
P denotes the pressure of the system.
V is the volume of the gas.
k is a constant value representative of the pressure and volume of the system
If you could help me in a direction to go that would be great thank you