If there are variations in pressure or elastic stress within the system,parts of the system may move, or expand or contract. Eventually these motions,expansions or contractions will cease, and when this has happened we say that the system is in mechanical equilibrium. This does not necessarily mean that the pressure is the same at all points. Consider a vertical column of fluid in the earth's gravitational field. The pressure in the fluid is in mechanical equilibrium under the influence of its own weight and an equal upward force arising from the pressure difference between its upper and lower surfaces.

This was written in my book (by Sears & Salinger). But (whatever anyone calls me) I could not understand what they wanted to tell about mechanical equilibrium. Can anyone please help to clarify what they talked about here?

[ Has mechanical equilibrium any relation with the surroundings of the system?]

  • $\begingroup$ I'm not clear on who 'they' are. But in the case of your column of water, there has to be container walls or else the fluid will find another equilibrium and lower energy level, flowing to the ground. So perhaps [they] were referring to the walls that kept the column of fluid in the shape of a column. The walls indeed part of the first state of equilibrium. $\endgroup$ – docscience Oct 13 '14 at 17:13
  • $\begingroup$ The passage is perfectly clear to me, so you may need to be more verbal about what you don't understand. The pressure gradient is balancing the weight of the fluid. $\endgroup$ – Alan Rominger Oct 13 '14 at 17:21
  • $\begingroup$ Ok, I want to know what they want to tell in this passage. What pressure are they talking about ; I am not understanding this. $\endgroup$ – user36790 Oct 13 '14 at 17:31
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    $\begingroup$ Imagine a column of water with walls on the sides bounding it, but also imagine there are movable horizontal partitions dividing the water, so you have a stack of water-filled chambers arranged vertically & each chamber has a moveable top and bottom. Then "mechanical equilibrium" would be reached when the movable partitions all settle on a fixed height, and the pressure not being uniform at equilibrium would mean the force exerted on a partition from below (force/area on its bottom side) could differ depend on which partition we were examining--is this a valid way of thinking of it? $\endgroup$ – Hypnosifl Oct 13 '14 at 18:07
  • $\begingroup$ @Hypnosifl : This seems to be better. So can you tell why should there be unequal force at different partitions at equilibrium? $\endgroup$ – user36790 Oct 13 '14 at 18:22

I don't think you need to overthink this so much. Mechanical equilibrium in this context basically means that from a macroscopic point of view, all forces are balanced; this usually also means that the system's parts are at rest, though a system in uniform motion could be considered in mechanical equilibrium, I guess.

The point that the authors are trying to make is that being in mechanical equilibrium (which remember, for our purposes means all net forces are zero and the system is at rest) doesn't necessarily imply that the pressure must be uniform over all the system.

As an example they tell us to consider a column of water inside some container. The water is subject to gravity, so if every small chunk of water is to be at rest, there must be some force counteracting gravity. And indeed there is; in fluids, internal forces come from pressure gradients. Therefore, if the pressure varies as $p = p_0 - \rho g z$ (where $p_0$ is some constant and $\rho$ the density; this is a well known formula in hydrostatics), then the pressure is higher at the bottom. This means that the internal forces, which are equal to minus the pressure gradient, point upwards and cancel out gravity.

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