Page:Elementary Text-book of Physics (Anthony, 1897).djvu/151

§ 118] the barrier. To set the siphon in operation it must be first filled with the liquid, after which a steady flow is maintained.

In this ease, as before, we may set $$\frac{A^2}{A_{1}^2} = 0$$, $$v_{1} = 0$$, $$p$$ and $$p_{1}$$ both $$= p_{a}$$, and $$(V_{1} - V) = gl$$, where $$l$$ is the distance between the surface level and the discharging orifice. The velocity becomes $$v = \sqrt{2gl}$$. The siphon, therefore, discharges more rapidly the greater the distance between the surface level and the orifice. It is manifest that the height of the bend in the tube cannot be greater than that at which atmospheric pressure would support the liquid.

The flow of a liquid into the vacuum formed in the tube of an ordinary pump may also be discussed by the same equation. The pump consists essentially of a tube, fitted near the bottom with a partition, in which is a valve opening upwards. In the tube slides a, tightly fitting piston, in which is a valve, also opening upwards. The piston is first driven down to the partition in the tube, and the enclosed air escapes through the valve in the piston. When the piston is raised, the liquid in which the lower end of the tube is immersed passes through the valve in the partition, rises in the tube and fills the space left behind the piston. When the piston is again lowered, the space above it is filled with the liquid, which is lifted out of the tube at the next up-stroke.

To determine the velocity of the liquid following the piston, we notice that in this case $$p_{1} = p_{a}$$ and $$p = 0$$ if the piston move upward very rapidly, $$(V_{1} - V) = -gh$$, where $$h$$ is the height of the top of the liquid column above the free surface in the reservoir, and $$\frac{A^2}{A_{1}^2}$$ again $$= 0$$. We then have $$\tfrac{1}{2}v^2 = \frac{p_{a}}{d} - gh.$$

The velocity when $$h= 0$$ is $$v = \sqrt{\frac{2p_{a}}{d}}\cdot$$ When $$h$$ is such that $$dgh = p_{a}$$, $$v= 0$$, which expresses the condition of equilibrium.

The equation $$v = \sqrt{\frac{2p_{a}}{a}}$$ expresses, more generally, the velocity