## Friday, December 19, 2014

### Fluid Statics Pressure and Density

Matter exists in three different states. If the force of attraction between the molecules is very strong the matter will have a particular shape which is strong and this kind of the State is called solid-state. As explained in the previous topics the substances in the solid-state exhibits elastic property.

If the force of attraction between the molecules is weak, the state is called liquid and if the force of attraction is very weak the State is called gas. Both liquids and gases has the property of flowing from one place to another place and that is the reason why they are called as fluids. Whenever there’s an external pressure difference, the fluids always close from one place to another place.

In the case of solids when an external pressure is applied the volume will reduce. But in the case of the fluids the force of attraction between the molecules is negligible and hence even when the pressure is applied there is not going to be any serious change in the volume. That’s why fluids in general in ideal stage are assumed as incompressible. So the liquids cannot support the shearing stress and they can easily slide one over the other.

The branch of the physics that deals with the fluids in the state of the rest and its applications is called as fluids statics. The branch of the physics that deals with the fluids in the state of motion and its applications is called fluid dynamics. Both in fluids statics and fluid dynamics the force of friction which is called as viscosity is  ignored. If the frictional force is also taken into consideration then the subject is called viscosity.

Density and pressure are the two important physical quantities that we learn in fluids statics. Density is defined as mass per unit volume.

Relative density or specific gravity is defined as the ratio of the density of the given substance to the density of the water at 4°C. Being it is the ratio of similar physical quantities it has no units and dimensions. 4°C is taken as the reference just because it 4°C volume of the water is minimum at the density is maximum.

We can calculate the density of the mixer of two different liquids into possible cases as shown below. In case one we are going to mix to liquids of different masses having different densities. It is very clear that if the two liquids which are going to be mix are having equal masses and when they are having equal volumes there is going to be different densities as shown below.

Pressure

Pressure is defined as the force acting on a body per unit surface area. It is being treated like a scalar quantity. Though it has both magnitude and direction it is not able to satisfy the vector laws of addition and hence it is being treated like a scalar quantity.

The atmosphere of the earth always applies a common pressure on all of us and it is called atmospheric pressure. It is being taken like a standard value and it is shown with one atmospheric pressure. Smaller the impact area larger than impact and it is measured as the pressure.

As per Pascal’s law pressure in a fluid at rest is same at all the points if all the points are at the same height. To prove this let us consider a tank containing a liquid. Let us assume two points separated by certain distance. Imagine a cylinder of the liquid about an axis around that height we can calculate the effective force and the pressure as shown below.

Determination of the atmospheric pressure using Barometer

The earth has atmosphere up to a height of 200 km from the surface of the earth. This causes the pressure on the bodies of the earth. This force is perpendicular to the surface of the body. Barometer is a device which can be used to measure this atmospheric pressure.

It consists of a pipe which is closed at one end and open at other end. It is filled with the mercury and placed gently in the mercury vessel as shown. The mercury in the tube falls down a little and finally stays at a certain position.

In the upper part of the closed tube  there is a vacuum and there is no air that is allowed to go to that place. Because of the vacuum there is no pressure developed at the point and pressure at the two points of the lower level is equal because they are at the same heights. Basing on this concept we can derive the expression for the atmospheric pressure as shown.

Determination of the pressure due to a gas

To determine the pressure due to a gas we needed different apparatus that is called as manometer. The apparatus consists of a vessel which is filled with a gas and we are going to find out the pressure due to that gas. It also consists of a U-shaped tube containing the same liquid. One end of the tube is open to atmosphere and other end is connected to the vessel as shown.

After the final adjustment,the liquid in the U-shaped but you which can identify two different points on the two sides which are at the same level and we can say being the levels are same the pressure is at the two points are same.

Pressure is independent of the amount of the fluid and the shape of the container. It is going to depend on the height of the liquid and when the shape of the container is different if the heights are same the pressure is going to be the same as shown below.

If the liquid is in a accelerator moving in the upward direction the pressure in the liquid is going to be different. We can derive the equation for the effect to pressure as shown below. The simple Newton laws of motion concept is taken into consideration while deriving this equations.

Pascal’s law

When the pressure is constant and the ratio of force per area is always going to be constant. Hydraulic lift works basing on this concept. It consists of two open parts with different areas of cross-section. One part will be having a smaller area of cross-section whereas the other part will be having a larger area of cross-section. On the second part where the larger area of cross-section is existing with can put a heavy body and we can lift that a body with a small force applied on the first part with a smaller area of cross-section.

In the following diagram we are even going to solve small problem on the same concept.

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