- #1
samsanof
- 5
- 0
Hi all,
In short: For an air leg or air spring, there is a method using a Taylor approximation to find the spring constant for very small displacements, but I can't seem to figure out how it works. I've learned that air legs don't follow Hooke's law very much at all, except for when the displacement is very small.
So to start: You have a vertical piston, with air underneath. It moves very slowly and the walls have high thermal conductivity, i.e. all processes are isothermal (and at 298 K, room temperature). The piston has area S and mass M, and there is a smaller mass on top, m (M>>m, so I expect amplitude of oscillations to be small). At rest, the piston is height h from the bottom of the cylinder and the air has pressure P. Then, mass m is removed, setting it into oscillations.
My attempt:
So I start out: Make a force balance, because initially it is at rest. the only variable is the height, so i put things in terms of h
In words: Pressure inside*area - mass*gravity - atmospheric pressure*area = 0, up-down=0
P*S - (m+M)g - Patm*S = 0 = Fnet = -kx for small x
The second two terms are constant.
Then, I think the next step is to say that the pressure inside is a function of h, rearranging pv=nrt and simplifying you get
n*R*T/h - (m+M)g - Patm*S = -kx
Now my though is, taking the derivative with respect to x yeilds k:
-n*r*T/(h^2) * h' = -k
Now I'm stuck. How does one go about finding h'? and where does the Taylor approximation come into play, for anyone familiar with that method?
As always, I appreciate all your efforts. Hopefully we can figure this out.
In short: For an air leg or air spring, there is a method using a Taylor approximation to find the spring constant for very small displacements, but I can't seem to figure out how it works. I've learned that air legs don't follow Hooke's law very much at all, except for when the displacement is very small.
So to start: You have a vertical piston, with air underneath. It moves very slowly and the walls have high thermal conductivity, i.e. all processes are isothermal (and at 298 K, room temperature). The piston has area S and mass M, and there is a smaller mass on top, m (M>>m, so I expect amplitude of oscillations to be small). At rest, the piston is height h from the bottom of the cylinder and the air has pressure P. Then, mass m is removed, setting it into oscillations.
My attempt:
So I start out: Make a force balance, because initially it is at rest. the only variable is the height, so i put things in terms of h
In words: Pressure inside*area - mass*gravity - atmospheric pressure*area = 0, up-down=0
P*S - (m+M)g - Patm*S = 0 = Fnet = -kx for small x
The second two terms are constant.
Then, I think the next step is to say that the pressure inside is a function of h, rearranging pv=nrt and simplifying you get
n*R*T/h - (m+M)g - Patm*S = -kx
Now my though is, taking the derivative with respect to x yeilds k:
-n*r*T/(h^2) * h' = -k
Now I'm stuck. How does one go about finding h'? and where does the Taylor approximation come into play, for anyone familiar with that method?
As always, I appreciate all your efforts. Hopefully we can figure this out.