# Questions on Pascal's Principle

I am aware that Pascal's principle says that if a pressure acts on an enclosed liquid, the pressure will be transmitted uniformly throughout the liquid.

• Can Pascal's Principle be applied to a manometer? When you apply pressure on one end, would that pressure be transmitted uniformly to the other end? When measuring difference in pressure with the manometer, is Pascal's Principle used? I mean the liquid is not really enclosed?

Normally, we apply this principle in a hydraulic press, where there are pistons which enclose the liquid, in a manometer, do you take the substance exerting the pressure (eg. gas being pumped) as the 'piston'?

• What does 'enclosed' actually mean? Liquid being covered on all sides with no air? Or being unable to flow freely?

• How is pressure transmitted if the liquid is not enclosed? (Eg.) A beaker.

The term enclosed is a little misleading. Look at the diagram below:

Both containers are of equal size and contain the same amounts of the same liquid. The left liquid is contained by a beaker and a weightless, frictionless piston, the right is contained by a beaker that is simply open to air. Both sit side by side on the surface of the Earth (and thus subject to gravity).

In both cases Pascal's Principle applies and the pressure in the liquids can be measured by means of a manometer.

If we assume the pressure of the surrounding air to be $p_0$ then in the right hand case the pressure in the liquid will only depend on the height $h$ due to the weight of the liquid column, according to:

$p=p_0+\rho gh$, with $\rho$ the density of the liquid and $g$ Earth's acceleration.

In the left hand liquid the situation is very similar but we have to take into account the extra pressure exerted by the piston, which is $p_p=\frac{F}{A}$ with $F$ the force applied to the piston and $A$ the surface area of the piston.

We then get:

$p=p_0+\frac{F}{A}+\rho gh$.

So for left and right $p$ would be the same if $F=0$, despite the right not looking particularly enclosed.

The meaning of 'enclosed' becomes more apparent when we drill a hole in either container and liquid is now free to flow out of it. At that point Pascal's Principle no longer holds exactly.

Pascal's Principle applies only to static situations, where liquid is not free to escape its container (enclosure, if you prefer). But that does not mean pressure could no longer be measured with a manometer, only that the measurement would no longer correspond exactly to Pascal's Principle.

In the case of common hydraulic devices like car lifts, springing a leak renders the device useless because it no longer obeys Pascal's Principle.

• Hi, would the pressure acting on the sides of the left beaker be equal to the atmospheric pressure and and the pressure exerted by the piston? If we say pressure is transmitted uniformly? – Cr4zyM4tt Sep 30 '15 at 14:29
• @Cr4zyM4tt: yes, that is what $p=p_0+\frac{F}{A}+\rho gh$ means. But note that the pressure on the sides varies with $h$, so the deeper part experiences slightly higher pressure. Often in hydraulic devices (car lifts e.g.) the pressure applied by the piston is much higher than the $p_0+\rho mg$ part, in which case we can say: $p=\frac{F}{A}$, approximately. – Gert Sep 30 '15 at 14:35
• Thanks, can I say that the external pressure throughout the sides are equal? Based on Pascal's Law – Cr4zyM4tt Sep 30 '15 at 15:18
• @Cr4zyM4tt: what we can say is that the forces that act on both sides of the wall that makes up the container (liquid/wall/air) are the same, otherwise the wall would move (Newton). On the outside the pressure $p_0$ acts inwards, on the inside $p$ acts outwards. But $p>p_0$ and the 'deficit' is made up by the strength of the container, otherwise the wall would move, i.e. break or rupture. – Gert Sep 30 '15 at 15:26
• Thanks Gert, I think I am confused about Pascal's law saying that pressure is transmitted uniformly throughout the liquid, what bdoes this mean? Is it pressure experienced by an object anywhere in the liquid is uniform? For example, for the beaker, is the pressure exerted on the sides the pressure transmitted by the piston and air? – Cr4zyM4tt Sep 30 '15 at 16:51