# What special feature does this kind of coupling have?

This diagram was in the 4th chapter of the book Heat and Thermodynamics, Zemansky and Dittman.

An amount of gas confined to a cylinder of thermally insulator material, here, is undergoing an adiabatic expansion. As the gas expands, the body is lifted.

My question is that why is the mechanism designed like that? I mean that almond shaped thing attached to the small gear. There must be an idea behind it, as the book says:

My own guess is that this kind of hanging the mass, guarantees a quasi-static process.

• It is not clear to me what exactly you are asking, the upright rod has teeth which mesh with the gear wheel, producing the rotation as the piston rises, allowing the weight to be moved both vertically and horizontally inwards. Why they need motion in two dimensions, I don't know without context of some kind. Where did find this illustration, can you say.
– user140606
Jan 15, 2017 at 13:07
• @Countto10 that special curved shape is the question. Why does it need to be like that?
– AHB
Jan 15, 2017 at 13:14
• Sorry, didn't read your revised post, but how moving the mass inwards helps to provide a quasi static process, I am sorry to say, I have no idea. I do think this is an engineering question, thus the (not from me) d/v.
– user140606
Jan 15, 2017 at 14:11
• Good old Prof. Schmidt, he was so very clever that now nobody else can figure out what the damn thing is for :)
– user140606
Jan 15, 2017 at 14:16
• @Countto10 I have updated my answer. Jan 16, 2017 at 11:45

As the piston moves up the pressure inside the cylinder decreases.
Thus the force exerted by the gas in the cylinder on the piston decreases.
What is needed to allow the piston to move up slowly is to apply a decreasing external force on the piston as the piston moves up such that the external force is slightly less than the force exerted by the gas on the piston.

That decreasing force is achieved using the cam, rack and pinion.
There is a lever with axle of the pinion/cam acting as the fulcrum.
As the rack moves up the cam rotates and reduces the distance between the line of action of the weight and the axle thus reducing the torque produced by the weight and hence the torque exerted by the pinion.
Hence the force on the rack/piston is reduced.

Update

Schematic diagram of apparatus

When the piston has moved a distance $x$ from the bottom of the cylinder the rack has also moved upwards a distance $x$.
At the same time the pinion and the cam have rotated through an angle $\theta$.

$x=a \theta$ where $a$ is the fixed radius of the pinion.

For an adiabatic expansion $PV^\gamma = {\rm constant} = \dfrac F A (Ax)^\gamma \Rightarrow F x^\gamma = \rm constant$

With the symbols as defined in the diagram $F = \dfrac b a mg \Rightarrow b\;\theta^\gamma = \rm constant$ and this is the equation for the shape of the cam.

• If we know the pressure of the gas as a function of its temperatue and volume, can we find the polar equation of that curve? Or the curve differs for every temperature?
– AHB
Jan 15, 2017 at 17:12
• I think this totally depends of the equation of state of the system. Increasing the mass would scale the torque only. We don't know whether the behavior of the system at temperatures agreed with this scaling.
– AHB
Jan 15, 2017 at 17:25
• @AHB I have updated my answer. Jan 16, 2017 at 11:46
• +1 That is very perceptive of you, and the connection between an adiabatic equation and the actual physical utilisation of it is very impressive. So in effect we have a self shortening lever. I love the drawing, it has a blueprint feel to it. My apologies to the good Professor. Although I would still not fly in an aircraft designed and constructed by physicists alone...
– user140606
Jan 16, 2017 at 11:58
• With a measurement, that constant can be found. Any because this is assumed to be an ideal gas, temperature increment only needs change of the hanging mass. That equation is a sharper-than-normal hyperbola which look like the real curve of the cam. Everything seems to be fine. Thanks.
– AHB
Jan 16, 2017 at 12:18