# Liquid Pressure Variation with Depth

## Contents

### Description of the Experiment

In this experiment, we study the density of four different liquids by taking into account that pressure variation with depth depends on it.

• Video: rtsp://elabmc.ist.utl.pt/scuba.sdp
• Laboratory: Basic in e-lab.ist.eu
• Control room: scuba

<swf height="550" width="480">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/Scuba.swf</swf>

# Experimental Apparatus

In this experiment, there are four acrylic tubes with a diameter of thirty millimiters and one meter long. Each tube is filled with a different liquid: distilled water, salt water, glycerin and vegetable oil. Inside each tube there is a bell with an air bubble that allows pressure to be measured through a flexible tube, which is attached to a pressure sensor located outside the liquid.

The change in volume can be estimated considering that each bell has a volume of approximately $$2 cm^3$$. The hose has a cross-section of $$1 mm$$ and a length of $$1 m$$, but it can easily be ignored.

The tubes are mounted vertically, and the four probes move simultaneously as established by the configuration chosen. The latter pauses for a second at each measuring point to allow the pressure to stabilize before measuring. The experiment will take longer if the user requests many points.

# Protocol

The user must define the maximum and minimum height, as well as the number of samples to take across the path. This means that he can choose the initial and final depth of the probe's motion and obtain the data (each liquid's) on the variation of the pressure as depth changes. Afterwards, the data can be fitted to the following equation, and from that, the density of the various liquids can be determined.

$p(h) = p_0 + \rho g h$

If multiple runs are made (with different starting and ending points), the experimental error will be lower.

The following table shows the four liquid's density accepted values.

Material Accepted density ($$\frac{kg}{m^3}$$)
Water $$1,00 \times 10 ^3$$
Glycerine $$1,26 \times 10 ^3$$
Salty water $$1,03 \times 10 ^3$$
Vegetable oil $$0,92 \times 10 ^3$$

# Theoretical Principles

The pressure exerted by a liquid is proportional to the weight of the fluid column, meaning that it depends not only on depth but also on density. This can be determined through the relation between pressure and depth. This relation can be expressed mathematically by:

$p = p_0 + \rho g h$

where $$p_0$$ represents the pressure at the liquid's surface and $$\rho = m/V$$ it's density, being g the local gravity acceleration and h the depth. Recalling Pascal's principle note that $$p_0$$ is evenly distributed through the whole liquid.