A submarine can float above the water surface or dive below the water surface by emptying or filling its ballast tank. A ballast tank is cavities on a hull (larboard and starboard) of a submarine that can be filled with air or water. The submarine moves forward by using propellers and move downward by using hydroplane which looks like the tail of the airplane which pushes it downward. On the other hand, when the submarine is going to the surface, the water inside the ballast tank has to be pushed outward by the pressurized air.
The working principle of a submarine agrees with Archimedes' principle. What does the Archimedes' principle state' You will study it in this article?
A. Hydrostatics
There are three states of mater, i.e.: solid, liquid, and gas. In solid, a matter retains a definite size and shape. In a liquid, a matter has a definite volume and takes the shape of its container. Liquid and gasses have an ability to flow. That is why both liquid and gasses are usually called fluid or the flowing substance. Our study or
fluid will be divided into two parts, i.e. hydrostatics which discusses the static fluid or fluid at rest and hydrodynamics which discusses fluid in
motion.
1. Density
Density (𝝆) of a material is defined as its mass (m) per unit volume (V). Mathematically, density is formulated as:
The SI unit of density is kg/m3; meanwhile, the cgs unit of density is g/
cm3.
2. Pressure
a. Definition of Pressure
Pressure is defined as
the force per unit of area.
If a force F
works perpendicularly to a surface with area A, we can write the pressure P as:
The SI unit of pressure is Pascal (Pa), where 1 Pa = 1N/m2. The pressure will increase if the force applied to a given area increases or the area of a given force is reduced.
See Figure 6.3 with the same magnitude of force, the pressure given by a needle is bigger than that by a finger on a balloon. The pressure, given by a finger only causes the balloon to undergo a little deformation., whereas the same magnitude of
force given by a needle is able to blast the ballon.
b. Atmospheric Pressure and Gauge Pressure
We are all used to work under the pressure called atmospheric pressure. This pressure is caused by the weight of the air above us. The value and unit of atmospheric pressure are:
Have you ever measured the air pressure in a car's tire? What do you find? Has the pressure inside the tire ever been lower than the outside air pressure? Let's examine this.
In some cases, we sometimes measure the pressure inside a certain object by calculating its difference with the outside air pressure (atmospheric pressure). Even the air pressure inside a flat tire will never be zero. A measurement of the pressure inside a flat tire will result in the value of the outside air.
To inflate a car's tire so that the pressure inside it reaches the value of 241 kPa, the pressure inside the tire must be higher than the outside air pressure in the amount of P = 241 kPa + Po = 302 kPa. The241 kPa pressure that will appear in a measurement is called gauge pressure (Pg) which is generally formulated by
Pg = P + Po
c. Pressure in a Static Fluid
We have discussed pressure as the result of the division between the working force and the surface area. We have also discussed the example of pressure comparison two equal forces acting on a different surface area (Figure 6.3). The example describes the pressure caused by mechanical force; how about the pressure caused by fluid?
The experiment result shows that
fluid pushes in all directions. Thus, somebody who dives under water will feel the water pressure in whole parts of his body.
A point of fluid's particle also undergoes pressure from all directions. If the fluid is at rest, the forces acting on that point will be equal in magnitude. The direction of the compression force is always perpendicular to its contact surfaces.
Now, we area going to calculate the pressure in the bottom of a container filled with a density ρ and cross section of A (Figure 6.3). The pressure at the bottom of the container is caused by the compression force of the outside air and the weight of the
fluid above it.
The top of the fluid which is in direct contact with the air will undergo a compression force of
Fbottom = Ftop + mg
The bottom of the container will undergo downward force in the amount of Ftop plus the weight of the fluid, i.e. w = m.g;
Pbottom
= P0 + ρhg ....... (6.4)
Pressure caused by a fluid at rest is called hydrostatic pressure. The hydrostatic pressure is proportional to the depth (h) and the density (P) of the fluid. Any point at the same level always has the same pressure, regardless of the shape of the container. Equation (6.4) does apply for any depths, not only the base of the container.
Example 6.1
A drum of 1 meter high is filled with kerosene ( ρ = 0,8 g/cm3 to 3/4 of the drum's height. If g = 10m/s2, find the pressure at the bottom of the drum.
Answer :
= 800 kg/
m3 ; h = 0,75 m
Pbottom =
ρgh =
P0 + 800 kg/
m3) (10m/s
2) (0,75m) = 6,01 N/
m2.
3. Pascal's Principle
As a fluid, the earth's atmosphere exerts pressure to all materials which are contact with it, including all the fluids on the earth's surface. The atmospheric pressure applied to a fluid is transmitted to every part of the fluid. This was stated by a French philosopher and scientist, Blaise Pascal (1623-1622). Pascal's principle or commonly known as Pascal's law states that:
Pressure applied to an enclosed fluid is transmitted equally to every part of the fluid.
On of the applications of Pascal's law is the hydraulic lift, as shown, If a force F1 strikes a piston with a small area A1, the pressure of the fluid will increase the amount of P = F1/A1. The increase of the pressure is then transferred to the cross-sectional area of a larges piston A2. Since the applied pressure is the same at both pistons, it applies:
F1/A1 = F2/A2
4. Archimedes' Principle
A sand miner finds it easier to lift a bucket of sand from a river when the bucket is still on the water. However, when the bucket is taken out of the water, it will become heavier. How can this happen?
If a body is immersed in a fluid, the fluid exerts an upward force on the body. The force makes the body as though it had less weight. This fact was firstly acknowledged by Archimedes, it is therefore known as Archimedes' principle, which states that:
When a body is immersed in a fluid, the fluid exerts an upward force (buoyant force) on the body whose magnitude is equal to the weight of the fluid displaced by the body.
Archimedes' principle is mathematically formulated as:
FA = ρfluid V g
where is a buoyant force, ρfluid is the density of the fluid, and Vt is the volume of immersed body.
There are three possible cases of the body in a fluid, i.e floating, barely floating (completely submerged), and sinking. By using Newton's first law and Archimedes' principle, we can determine in what terms a body will be floating, barely floating, or sinking.
a) Floating
A body is said to be floating if only some parts of the body are submerged or beneath the water surface, meanwhile the other part is above the water surface. On a floating body, the magnitude of buoyant force FA is equal
FA = mg
ρfluid g Vt =
ρb Vb g
ρfluid Vt =
ρb Vb
The volume of the body below the water surface Vt is always smaller than the total volume of the body Vb. Hence he density (ρb) of a floating body is smaller than the fluid's density ρb < ρfluid
b) Barely Floaing (Completely Submerged)
On a barely floating body, the magnitude of buoyant force
FA is equal to the body's weight w = mg, Thus, we have:
ρfluid g Vt =
ρb Vb g
However, the volume of the immersed body
Vt is equal to the total volume of the body
Vtb . Hence for a barely floating body, we have the relation
ρb =
ρfluid . Therefore on a barely floating body, the density of the body is equal to the density of fluid.
c) Sinking
When a body is sinking, it means that the buoyant force
FA acts on the body is smaller than its weight
mg. On a sinking body, the volume of the immersed body
Vt is equal to the total volume of the body
Vb. However, if the body stays at the bottom of a container, there will be a normal force N acting on it, so it goes:
FA + N = w
N =
ρb Vb g -
ρfluid g Vt
Since the normal force is always positive, we have
ρb <
ρfluid . Thus, a body sinks in a fluid when its density is greater than the density of the fluid.
Archimedes' principle is widely used in either technology or science. Some examples of devices that work based on Archimedes' principle are hydrometers, ships, dockyards, submarines, and hot air ballons.
5) Viscosity
When sliding across a rough floor, a block experiences a frictional force opposing then motion. Similarly, a fluid flowing past a stationary surface experiences a force opposing the flow. This tendency to resist flow is referred to the viscosity of a fluid. Fluid like air have low viscosity, thicker fluid like water are more viscous, and fluids like honey and molasses are characterized by their high viscosity.
A fluid's viscosity is related to the frictional force between fluid's layers when a layer slides over him other. On liquid, the viscosity is caused mainly by the cohesive force between its molecules; meanwhile, on gas, the viscosity is caused mainly by
collisions between its molecules.
The ideal fluid we often use in the discussion has zero viscosity. This is because we neglect the inter-molecule interaction of the ideal fluid. The flow of ideal fluid is laminar (streamline), every point in the fluid has an equal speed.
the true fluid with the certain value of viscosity will generate a flow pattern as. Because adjacent portions of the flow pasts one another with different speed, a force mus be exerted on the fluid to maintain the flow.
The force causing a viscous fluid to flow is provided difference by the pressure, (P1 - P2) across a given length of tube (L). Experiments show that the required pressure difference is proportional to the length of the tube (L) and the average speed of the fluid (v), and inversely proportional to the cross-sectional area of the tube (A).
We can observe viscosity by dropping a marble into a glass of cooking oil. The marble is slowed down by the fluid due the frictional force. The magnitude of the frictional force on the marble moving at its constant velocity
v in the fluid can be calculated by using the following equation:
f = 5πηrv
where is fluid's viscosity and r is the marble's radius.
When the marble is dropped into the oil, at a certain time it will move in a constant velocity. The velocity is known as terminal velocity. There are three kinds of force acting on the marble during its movement on the cooking oil,i.e. weight (
w) buoyant force (
FA), and frictional force of fluid (
f).
