DENSITY ANALYSIS OF A SYSTEM

ARTICLE INFORMATION

Writer Muhammad Tayyab

Registration no 2017-ME-110

Supervisor Dr Jawad Sarwar

INTRODUCTION

As we know that our major task is to determine the density of water filled in any container or pot , the density of the pot and the combined density called bulk density of the system.

HISTORY

ABSTRACT

Density is the most important parameter in determining the properties of a system. Our major task is to determine the density of water and the density of the pot and the combined system under general circumstances

PROBLEM STATEMENT

This step involves the process of explaining the problem statement that what we have to find and what is given to us in the statement of the problem. In the problem assigned to us we have to calculate the density of the water and the pot in which the water has been filled and combined density `

Sometime Around 250 BC. The Greek Mathematician Archimedes Was Given The Task Of Determining Whether A Craftsman Had Defrauded The King Of Syracuse By Replacing Some Of The Gold In The King’s Crown With Silver. Archimedes Thought About The Problem While Relaxing In A Bathing Pool. As He Entered The Pool, He Noticed That Water Spilled Over The Sides Of The Pool. Archimedes Had A Moment Of Epiphany. He Realized That The Amount Of Water That Spilled Was Equal In Volume To The Space That His Body Occupied. This Fact Suddenly Provided Him With A Method For Differentiating A Mixed Silver And Gold Crown From A Pure Gold Crown. Because A Measure Of Silver Occupies More Space Than An Equivalent Measure Of Gold, Archimedes Placed The Craftsman’s Crown And A Pure Gold Crown Of Equivalent Mass In Two Tubs Of Water. He Found That More Water Spilled Over The Sides Of The Tub When The Craftsman’s Crown Was Submerged. It Turned Out That The Craftsman Had Been Defrauding The King! Legend Has It That Archimedes Was So Excited About His Discovery That He Ran Naked Through The Streets Of Sicily Shouting “Eureka! Eureka!” (The Greek Word For “I Have Found It!”).

METHODOLOGY TO TACKLE THE PROBLEM

We will use problem solving technique to provide an efficient solution to the given task. This technique involves a series of steps to be followed to reach a solution that includes

Problem statement

Schematic

Assumptions and approximations

Physical laws

Properties

Calculations

Reasoning, verification and discussion

These steps are to be followed to be followed before solving the problem

SCHEMATIC

In this step we have to give the 2d view or as we called in layman language diagram of the system and label its details in this experiment I have used a steel pot as shown in the figure

Figure 1 steel vessel

The above steel vessel was used for the experiment and it had the following dimensions

DO=134.894mm=0.135m

DI=132.63mm=0.132m

thickness = t=1.132mm=0.001132m

Height=162mm=0.162m

mass of steel pot 803g=0.803kg

ASSUMPTION AND APPROXIMATIONS

We make assumptions and approximation for the ease in calculation and basically assumption is not based on truth but it is the best possible prediction of anything and approximation are based on the fact of similarity but due to these two factors the readings may vary but calculation becomes easy. In our this task we also may assumptions such as the vessel is ideally cylindrical and its is purely made up of steel and the water we filled in the vessel ids pure but in reality tap water was used and the steel vessel was not complete cylindrical but curved form the edges and the mass of knobs at the diameter of the vessel was ignored which due to its small mass will not significantly affect the actual value.

PHYSICAL LAWS

A physical law or a law of physics is a statement “inferred from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present. Physical laws are typically conclusions based on repeated scientific experiments and observations over many years and which have become accepted universally within the scientific community. The production of a summary description of our environment in the form of such laws is a fundamental aim of science. In this experiment we have used Archimedes law of density which states that mass per unit volume is called density and its equation is as

?=m/V

Archimedes’ Principle says that the apparent weight of an object immersed in a liquid decreases by an amount equal to the weight of the volume of the liquid that it displaces. Since 1 mL of water has a mass almost exactly equal to 1g, if the object is immersed in water, the difference between the two masses (in grams) will equal (almost exactly) the volume (in mL) of the object weighed. Knowing the mass and the volume of an object allows us to calculate the density.

Figure 2 measuring cylinder

PROPERTIES

In this part of problem solving technique we have to find unknown states with the help of the present state of the system. For example in this case we don’t have the volume of the vessel which is the unknown state but we do have the dimensional properties like height, diameter and thickness of the vessel so these are called known states and from these known states we can evaluate the volume of vessel.

V=?/4 (?Do?^2-?Di?^2 )+?Do×t

By putting the values of known parameters we get

Vsv=1.0060×10-4m3

Volume of water as measured by measuring cylinder is 1000ml=0.001m3

Vw=0.001m3

Mass of water =0.989kg

Mass of steel vessel=0.803kg

CALCULATIONS

This step involves applying the physical laws to get the desired quantity. In this case as we have the values of volumes of the components of the system so we will apply the law of density to get the value of density

?=m/V

WATER

As we will first calculate it for the water

?=0.989/0.001 kgm^(-3)

?=989kgm^(-3)

STEEL VESSEL

Calculations for steel vessel

?=0.803/(1.0060×?10?^(-4) ) kgm^(-3)

?=7962kgm^(-3)

BULK DENSITY

In this case we have to sum up the volumes and the masses to find the bulk density. Mass of the system is the sum of the mass of the water and vessel and volume of the system is the sum of the volume of the vessel and the water

Let m1 is the mass of the water and m2 is the mass of the vessel than mass of the system is as

m_s=m_1+m_2

m_s=0.989+0.803

m_s=1.792kg

Similarly v1 is the volume of the water and v2 is the volume of the vessel so volume of the system is

V_s=V_1+V_2

V_s=0.001+1.0060×?10?^(-4) m^3

V_s=0.0011006m^3

So as we can see that we have both the volume and the mass of the system so we can calculate the density of the system as by the formula of the density

?=m/V

?=1.792/0.0011006 kgm^(-3)

?=1628kgm^(-3)

Hence the bulk density of the system is 1628kgm-3

REASONING, VERIFICATION AND DISCUSSION

In this part of the problem solving technique we discuss the reasons for the deviation of the experimental results from actual results and verify them. In this case we have the density of the water as 989kgm-3 but actual value of the density is 1000kgm-3. So we will discuss factors that affect the density at any instant

DEVIATION IN THE DENSITY OF WATER

Temperature and salinity both affect the density of water. As the temperature of the water decreases, the density of the water increases. When water cools, it causes the molecules to move closer together as they slow down. This slowdown of the molecules causes the water to become denser. The opposite happens as water heats up. As the temperature of the water rises, the molecules speed up and begin moving farther away from one another. As the molecules begin spreading out, the water becomes less dense. On average, the less dense the water, the higher the level it floats within bodies of water. For instance, the bottom of the ocean contains water that is much denser than the water on the top. Salinity also plays a part in the density of water. Water that contains salt is much denser than water that does not contain salt. The more salt in the water, the denser it becomes Scientists define density as the measure of mass per volume of liquid. The equation for calculating density is d = m / v. Rearranging the equation can help users determine the mass when given the density. This equation is mass = density x volume.

Figure 3 graph effect of presence of salt in water on density

From the above graph we can see the relation between the density and salinity of the fluid and we will see another graph showing relation between the density of the water and the temperature. First due arise in temperature the density increases because on melting the volume decreases and density increase but after then onwards trend shifts to opposite

Figure 4 graph for relation between temperature and density for water

DEVIATION IN THE DENSITY OF STEEL

The value of density of steel is 8050kgm-3 and the experimental value is 7962kgm-3. The main reason for this error is the impurity of the substance. The variation of the density from the actual density is related directly to the nature of the impurity present in the steel if it is heavier than the steel than density will increase but if it is lighter than the steel then the density of the steel will decrease.

ACKNOWLEDGEMENTS

I am heartily thankful to my supervisor, Dr Jawad Sarwar whose encouragement, guidance and support from the initial to the final level enabled me to develop an understanding of the subject.

Lastly, I offer my regards and blessings to all of those who supported me in any respect during the completion of the project.

MUHAMMAD TAYYAB

REFERENCES

Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, Inc., ISBN 0-07-015000-1

Massey, B.; Ward-Smith, J. (2005), Mechanics of Fluids (8th ed.), Taylor & Francis, ISBN 978-0-415-36206-1

White, Frank M. (2003), Fluid Mechanics, McGraw–Hill, ISBN 0-07-240217-2

Nazarenko, Sergey (2014), Fluid Dynamics via Examples and Solutions, CRC Press