Fluids Unlike a solid, a fluid can flow. Fluids conform to the shape of the container in which it is put. Liquids are fluids the volume of which does not change usually (needs a large force). Gases are fluids which can change the volume easily with a small force. For all fluids, there is no tangential force ( a fluid flows because it cannot withstand a tangential stress). However, it exerts a force in the direction perpendicular to its surface. For fluids, we normally describe with DENSITY and Pressure instead of mass and force. To find the density of a fluid at any point, we isolate a small volume element round that point, and measure the mass of the fluid contained in that element. Then Density (unit :kg m -3 ) If the fluid is uniform (or smooth) then

Pressure The pressure at a given point in a fluid can be obtained by choosing a small area element and measuring the force acting perpendicular to it Pressure (unit: N m -2 : pascal : Pa) If the force is uniform over a flat surface, we use For a fluid at rest, the pressure p at a given point in a fluid has the same value in all directions. Hence pressure is a scalar. Pressure of atmosphere, normal condition= 1.01x10 5 Pa.

Fluids at rest Figure shows a tank of water, open to the atmosphere. We want to compare the pressure at two levels 1 and 2.

Pressure in a fluid To find the pressure p in a liquid at a depth of h below the liquid surface, choose level 1 to be the surface and level 2 at a depth h. The pressure at a point in a fluid in static equilibrium depends on the depth only but not on the horizontal dimension of the fluid or container. Note is the atmospheric pressure and is the total (absolute) pressure at level 2.

Measuring Pressure Figure a shows a basic Mercury barometer, a device used to measure the pressure of the atmosphere. The long glass tube is filled with mercury and inverted with its open end in a dish of mercury. The space above contains only mercury vapour with negligible pressure. Figure b is a more fanciful version

Measuring Pressure An open-end manometer measures the gauge pressure p g of a gas. It consists of a U- tube containing a liquid, with one end connected to the vessel whose pressure we want to measure, and the other. We can measure the pressure in terms of the height h as shown.

Pressure Example 1 The U-tube as shown contains two liquids in static equilibrium: water of density 998 kg m -3 is in the right arm and oil of unknown density is in the left. Measurements give l=135 mn and d=12.3 mn. What is the density of the liquid?

Pascal’s Principle A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container. Consider the case of the liquid in the tall cylinder, fitted with a piston on which a container with lead shots rests.

Pascal’s Principle and the Hydraulic Lever

Archimedes’ Principle Figure a shows a thin sac filled with water. The downward gravitational force on the contained water is balanced by an upthrust from the surrounding water. This buoyant force exists because the pressure from water below is greater than the pressure near the top. If we replace the sac with an object of exactly the same shape, the object displaces the water. The buoyant force is still the same because the forces at the surface are as before. The object may rise or sink depending on its weight (density).

Archimedes’ Principle For a floating body, the magnitude of the buoyant force is equal to the weight of body. When a body is heavier and is totally submerged, the buoyant force is enough only to balance part of the weight When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the body.