## Chapter 13: Liquids

The atoms or molecules in liquids have a strong attraction for each other, but the forces holding them together do not hold them in fixed positions relative to each other, but rather they can "slide past" each other and easily change the shape of the liquid.

### Pressure

• An important concept in both liquids and gasses is pressure. Pressure is defined as the force per unit area. A force of 100N spread over 0.5m2 produces a pressure of 200N/m2 or 200Pa. (1Pa = 1 Pascal = 1N/m2.) Similarly a pressure of 1,000Pa acting on an area of 3m2 will produce a force of 3,000N.

• Gravity acting on a liquid will tend to cause the liquid to deform so that the liquid is as close to the center of the earth as possible. If the liquid is confined by the walls of a container, i.e. jar, the liquid will deform to make the surface of the liquid as close to the center of the earth as possible. (Note that other forces can alter this shape slightly – e.g. adhesion of the liquid atoms or molecules to the surfaces of the container or the cohesive forces holding the liquid atoms or molecules together will change the shape of the liquid surface.)

• Gravity acting on a static liquid will produce a pressure in the liquid. This pressure will increase as you move down in the liquid. In the container at the right, the pressure is greater at B than at A. However, the pressure at C is the same as the pressure at B, i.e. the pressure will not change as you move horizontally, only vertically. (Note that this is only strictly true in a static liquid.) If the vertical distance between A and B is h, and the density of the fluid is D, the difference in pressure between points A and B is

PB – PA = Dgh

Where g = 10m/s2 on the earth. If the liquid is water, D = 1,000kg/m3, and the height difference is 2m, the pressure at B is 20,000N/m2, or 20,000Pa, greater than the pressure at A.

One way of understanding this is to note that the fluid at B must support the weight of the material above it, and there is more material above B than above A.

### Buoyancy

• If an object is completely submerged in a liquid, it displaces, or pushes aside, a volume of liquid equal to its own volume. If it is only partially submerged, the volume of liquid is less than it own volume.
• If I weigh an object in air with a scale and weigh it when it is immersed in liquid, e.g. water, the "weights" will not be the same. It will appear to weigh less in water. Actually its weight will not change, but the water exerts a force on the material so that the springs in the scale will not have to support the entire weight of the object. (It is like weighing yourself while leaning on a counter. The scale will not read your true weight because the counter is supporting part of your weight.) This force the liquid exerts on the object is called a buoyant force.

• The buoyant force on an object is equal to the weight of the liquid displaced. One way of looking at it is to note that the pressure at the top of the object pushing it down is less than the pressure at the bottom pushing it up, resulting in an upward force, the buoyant force.
• If an object is submerged in a liquid and weighs more than the buoyant force on it, it will sink, if only gravity and the buoyant force act on it. If it weighs less than the buoyant force it will rise and float. If an object floats, it weighs the same as the liquid it displaces.
• Note that if a solid object sinks, it displaces a volume of liquid equal to its own volume.

## Pascal's Principle

Pascal's principle says that if I increase the pressure by Pinc on one part of a liquid in an enclosed container, the pressure everywhere in the liquid will increase by the same amount, Pinc.  In the figure at the right, the pressure under the piston, i.e. the plunger, was simply the air pressure, 105Pa..  At the bottom of the container it was 1.5x105 Pa because of the variation of pressure with depth.  When I push on the piston, of area 0.01m2, with a force of 200N, I increase the pressure under the piston by Pinc = Force/Area = 2x104Pa.  Then the pressure just under the piston becomes P = 105Pa + Pinc = 1.2x105Pa.  The pressure at the bottom will also increase by 2x104Pa and become 1.7x105Pa.

This effect is used in hydraulic systems (power steering, brakes etc.)  It allows us to amplify force, but not work!  The typical arrangement is to have two pistons of different areas as shown at the right.  The left piston has an area of 0.01m2 and the right one and area of 0.05m2.  If I place a 5kg mass (weight = 50N) on the left piston, the pressure will increase by 5x103Pa.  It will also increase by 5x103Pa under the right one.  This produces an extra force on the right piston of 5x103Pa x 0.05m2 = 250N.  Note that for the two forces to "balance", Fleft/Aleft = Fright/Aright , i.e. the two pressure changes must be equal.