# Relative Density report

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Experiment 10: Relative Density Angelie Alexie C. Nethercott Department of Mathematics and Physics College of Science, University of Santo Tomas

España, Manila Philippines

Abstract

1. Introduction

Archimedes's tale took place some 2,200 years ago when King Hieron II of Syracuse suspected that the jeweller had substituted some of the gold for cheaper metal like silver so he asked Archimedes to prove his suspicion. Archimedes had spent a long time trying to figure out the answer, which came to him when he noticed how water would splash out of his bath tub the moment he stepped into it, and the more he stepped into the tub, even more water got displaced.

So to find the crown’s volume, Archimedes had to do was essentially immerse the crown and exact measurement of pure gold in a tub filled with water to the brim, measure the spillage, and compare the volume of spillages if the jeweller had indeed made a crown of pure gold the volume should be the same.

Archimedes' soak in the tub gave rise to Archimedes Principle, which states that when a body is immersed in water, it experiences a kind of force we call buoyancy. This force is equal to the weight of the water displaced by the body.

Buoyancy explains why something floats, and others don't. For instance, a ball of steel will sink because it's unable to displace water that equals its weight. But steel of the same weight but shaped as a bowl will float because the weight gets

distributed over a larger area and the steel displaces water equal to its weight.

Relative density, or specific gravity, is the ratio of the density (mass of a unit volume) of a substance to the density of a given reference material. Specific gravity usually means relative density with respect to water. The term "relative density" is often preferred in modern scientific usage. 

Nowadays, density becomes very important for example, ships require ballast to stay upright in the water, airplanes use counterweights to ensure they fly correctly. In either case, during the initial design, engineers must account for how much weight they need, and how much space must be allotted for it. To determine how much space they need, they must know the density of the materials they plan on using.

The purpose of this experiment was to determine the composition of a substance based on its density and by Archimedes principle.

2. Theory Density

The density of a material is defined as the ratio of its mass m to its volume V. Mathematically,

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(eq. 2.1)

The density of the material is an intrinsic property. Meaning, it does not depend on the amount of material. On the contrary, both of its mass and volume are extrinsic properties; that is, it is dependent on the amount of the material.

There are three (3) methods that can be used in obtaining the density of the material. First, it is the direct method. The direct method is used for regularly-shaped objects. For instance, calculating the volume of a cubic object, say sugar cube, is fairly straightforward as its volume is equivalent to the cube of its sides s(Vsquare= s3). Using a

balance, one can obtain its mass.

The second method is the indirect method. The indirect method is used when the material is irregular in shape. For instance, measuring the volume of a piece of rock using geometric formulas is intensely hard as you will measure every side of it. In this method, the material is suspended in a liquid and recording how much the liquid displaced. The portion of the liquid that is displaced is approximately equal to its volume.

The third method is based from the Archimedes’ Principle. In this method, a given material will be suspended in a liquid that has approximately the same density.

Relative Density

The relative density (RD) is defined as the ratio of the density of material with respect to some reference standard. In most

general sense, the standard is water having a = 1.00g/cc. Mathematically,

(eq. 2.2)

Because both have the same units, the relative density RD of a material is unit less.

The specific gravity (SG) of a material is equivalent to the relative density with a reference standard of water. Since the density of water is 1.00g/cc, the density of the material is equivalent to RD in units of g/cc. Mathematically,

(eq. 2.3)

When an object is in a fluid, there is buoyant force acting on the object due to the pressure of the fluid. The Archimedes Principle states that the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the object. Mathematically,

(eq. 2.4) Where w is the weight of an object,

= weight of the water, = weight

of the object in air, = weight of the object submerged in water.

3. Methodology

The experiment used electronic gram balance, spring balance, 100 ml. graduated cylinder and a beaker.

For Activity 1, a piece of brass was weighed using the electronic gram balance. Some water was placed in a graduated cylinder and noted the initial

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level of the water. The brass was placed in the cylinder and because of impenetrability; the brass displaced a volume of water equal to their volume and the new level was noted. The volume of brass was obtained from the difference of the two levels of water. The density was also determined using equation 2.1

Figure 1.0: Experimental set-up of displacement method (Photo credit: middleschoolchemistry.com)

For Activity 2, the weight of the bone in the air was obtained by using the spring balance. In a beaker with water, the bone was immersed and obtained its weight in water. Use equation 2.4 to compute for relative density (R.D.) and for density of the bone, multiply the relative density to the density of water (ρbone = R.D.* ρwater).

4. Results and Discussion

Table 1: Displacement Method for Alloy

Weight of the brass (g) 52.92

Initial level of water (cc) 60 Final level of water (cc) 67 Volume of brass (cc) 7 Density of brass (g/cc) 7.56 Experimental % by weight brass

11.76%

(Table 1 shows the result of the volume and density of the brass using displacement method)

Given a theoretical density of a brass which is 8.5 g/cm3, an 11.76% composition was obtained.

Table 2: Density of a Bone Weight of the bone in air (g)

80 Weight of the bone in water (g)

40 Relative density of bone 2 g/mL Density of bone (g/cc) 2

Finding Osteopetrosis

(Table 2 shows the results obtained in measuring the density of the bone and the relative density using equation 2.4)

The World Health Organization (WHO) defines osteoporosis on this scale :

Osteoporosis 2.5 Osteopetrosis 1 Normal Osteopenia

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After obtaining 2 g/cc as the density of the bone, it is found out that the bone has an Osteopetrosis because it is higher than the normal bone density which is 1.

5. Conclusion

After performing the experiment, it was evident that the force present which the fluid exerts on an object placed in it is equal to the weight of the fluid object displaces. Archimedes’ principle also makes possible the determination of the density of an object, that its volume cannot be measured directly. If the object is weighed first in air and then in water, the difference in weights will equal the weight of the volume of the water displaced, which is the same as the volume of the object.

6. Application

1. How can you distinguish “Fool’s Gold” from pure gold?

If they are both to be suspended in water, fool’s gold tend to float while pure gold will sink to the bottom because the density of pure gold is around 19.3g/cc while that of fool’s gold is only about 5.02g/cc.

2. The solid samples used in the experiment are denser than water. How will you determine the density of a solid that is less dense than water? Explain the formula that you will use.

First, put the solid in the water. Note how much water is displaced by the floating solid. Afterwards, submerge the solid fully under the water. Note again the water displaced. Calculate the relative density (RD) by dividing the density of the solid (floating) by when it is submerged. Since the density of water is 1.00g/cc, then the RD is equivalent to the density of the solid itself.

3. The suitability of a person to donate blood may be tested by placing a drop of his blood in a saline solution of density 1.03g/cc. Is he a suitable donor if the drop of blood sinks? Explain your answer.

The normal blood density is 1.06g/cc. If it sinks in the saline solution, the density will be somehow closer to 1.06g/cc. He is a suitable donor then.

4. What is the meaning of the expression “tip of the iceberg”? Is there a physical basis for this?

The expression “tip of the iceberg” is used to refer to a situation in which you or someone else is seeing only a portion of it. The physical basis for this is that when you see icebergs, you are only seeing a portion of its entire body because the majority are submerged underwater.

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5. Normal relative density of urine is from 1.015-1.030. What might be said if during urinalysis, a specific gravity higher than normal is obtained.

If someone has urine with a specific gravity higher than the normal range, it means that he/she has a more concentrated urine. During urinalysis, it may be said that the person is severely dehydrated.

7. References

http://www.princeton.edu/~achaney/tmve /wiki100k/docs/Relative_density.html  L. Marder. 1972. Vector Fields. George Allen & Unwin Ltd, London.

 H. Lass. Vector and Tensor Analysis. Phoenix Press, Quezon City, Philippines.  MIT OpenCourseware. Cartesian Coordinates and Vectors. Obtained July 8, 2013 from the MIT OpenCourseware site.

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