E206: Archimedes' Principle

ARCHIMEDES’ PRINCIPLE Bundalian, Patrick John Edbert G., Phy11L/A3 [email protected]

Abstract Archimedes Principle states any fluid applies a buoyant force to an object that is partially or completely immersed in it: the magnitude of the buoyant force equals the weight of the fluid that the object displaces. In performing the first part of the experiment what we did was we recorded the weight in air and the weight in water of the two solids. Proceeding to the second part of the experiment what we did was we chose one of the two metals and we recorded its weight while submerged into the two unknown liquids. For the third part of the experiment what we did was measure the density of the two unknown liquids with the use of a hydrometer. For the last part of the experiment what we did was we first recorded the weight in air of the cork, then the weight of cork in air and the sinker in water. Keywords: Buoyancy, Buoyant force, Depth, Dense object, Hydrometer

Introduction Archimedes discovered that the weight of a body in air minus its weight in liquid is equivalent to the weight of the liquid displaced by the body. When a body or an object is fully or partly submerged in a liquid, that body experiences an upward force called buoyant force. Also the displaced liquid is the volume of liquid equal to the volume of the body below the waters’ surface. Density is a characteristic physical property of a substance which means that there are no two materials have the same density. Specific gravity is defined as the weight of the body compared with an equal amount of pure water at 4OC wherein water is densest. The buoyant force is described by Archimedes’ principle as: an object, when placed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. The principle applies to an object either entirely or partially submerged in the fluid. The magnitude of the buoyant force depends only on the weight of the displaced fluid, and not on the object’s weight. Using Archimedes’ principle, you can deduce that an object: 1. Will float in a fluid if the object’s density is less than the fluid’s density (POPf). 3. Will remain in equilibrium at a given submerged depth if the object’s density is exactly equal to the fluid’s density at that depth (Po=Pf). The buoyant force on a floating object Fb is related to the properties of the displaced fluid by: Fb = mfg = pfVog

(1)

Where pf is the density of the fluid, Vo is the volume of the submerged part of the object, is the acceleration due to gravity, and mf is the mass of the floating object. The volume of the submerged part of a cylinder oriented vertically is equal to its cross-sectional area A multiplied by the height of the submerged part, so the buoyant force on it is: 1

Fb= mfg = pfAgh

(2)

This is a linear relationship between Fb and , so if you lower the cylinder into a fluid as you measure its weight, then plot Fb vs. h , the slope of the plotted straight line will be pfAg , i.e., directly proportional to the density of the fluid. This is a cool way to determine the density of an unknown fluid. You can determine the density of an unknown solid object in a similar fashion. It’s easy to measure the mass of an object, but unless it has a regular shape it’s not so easy to measure its volume. But Archimedes showed us how to measure volume by measuring weight. When the object is completely submerged in water, its weight (but not its mass) will decrease by an amount equal to the upward buoyant force the water exerts on it. So, ∆Wo = WA - Ww

(3)

Where ∆Wo is the loss of weight of water, WA is the weight of an object in air and W w is the weight of an object in water. This upward force is also equal to the weight of the displaced water. Or, ∆Wo = Ww = mwg = pwgVw

(4)

Where mw is the mass of an object in water and pw is the density of water. But the volume of the water is equal to the volume of the object. So, Vw = Vo =

∆Wo pwg

(5)

Therefore, the density of the object is, Po =

Mo Vo

Mo pwg =

∆Wo

(6)

You can also determine the density of an unknown liquid without measuring the submerged height of the solid object. With an object with density greater than that of the unknown liquid, first weigh it in air, then when it is submerged in the liquid, and then when it is submerged in water. By an analysis identical to that for the density of a solid object, you can show that, ∆Wo(inliquid)

Po = ∆Wo(inwater)Pw

(7)

Specific gravity is defined as the weight of the body compared with an equal amount of pure water at 4ºC (4ºC is the temperature at which water is densest). It also tells the number of times a certain material is denser than water. Specific gravity has no unit. The specific gravity of a substance is the ratio of that substance to the density of water. Mathematically: ps

SGS = pw

2

(8)

Where SGs is the specific gravity of a substance, PS is the density of the substance and pw is the density of water.

Methodology In this experiment, we used a digital balance – a simple two-button operation and visual menu prompts that allow students to begin weighing with minimal instruction; a piece of hydrometer –that has an ability to find the density of various fluids by putting the float and chain into the fluid, and measuring the amount of chain which floats; two pieces of 250-ml graduated cylinders – glassware that can hold liquids; three pieces of 250-ml beaker; one piece of cork, string and metal specimen. The first part of the experiment deals with the determination of the specific gravity of an unknown solid sample heavier than water. Where, the first metal sample (the gold one) at one side of a platform balance was suspended and found its weight in air (WA). Afterwards, we submerge the sample completely in a beaker of water and measure its weight while it is in water (Ww). We computed for the loss of weight of the sample using (eq. 3). Additionally, the specific gravity is also determined using the equation: G = WA / WA- Ww. We repeat the same procedures using the other sample (the white one) and compared the experimental value with the actual values. We identified that sample 1 was brass and sample 2 was a aluminum. Moving on to the second part, which is the determination of the specific gravity of an unknown liquid sample. We choose the aluminum as our metal sample to be used again in this part. We adjusted the string that is slightly tied up on the hook in such a way that the aluminum would be submerged completely in the first liquid sample and recorded its weight in liquid. Again, using (eq. 3), we find the loss of weight of body in liquid and determined the specific gravity using the equation: SG = WA-WL / WA-WW. Following the same procedures, we changed the liquid sample, compared the experimental with the actual values and finally identified the liquid samples. Before proceeding in the third part, we make sure that the liquids were transferred into two separate thoroughly dried graduated cylinders. The results gathered from the second part can be seen using another apparatus which is by a hydrometer. Whereas, it is placed inside the graduated cylinder, letting it float and record the reading. A higher specific gravity will result in a greater length of the stem above the surface while lower specific gravity will cause the hydrometer to float lower. Completing the whole experiment, which is the determination of specific gravity of a solid lighter than water; for this, the cork’s weight was recorded and being able to suspend from a string together with the sinker we’ve chosen (brass). We find the weight with just the sinker underwater, WCA-SW , and with both sinker and cork underwater, W(S+C)w . Considering the given weights, we compute for the loss of weight of cork using (eq. 3). Lastly, we determined the specific gravity of the cork using the equation: SG = WA / WCA-SW - W(S+C)w .

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Results and Discussion

(a)

(b)

(c)

(e)

(d)

Fig 1. The equipment used in the experiment (L-R): (a) electronic weighing scale and iron stand; (b) unknown metals and cork; (c) beaker; (d) hydrometer; (e) graduated cylinders. In the first part of the experiment, the specific gravity was determined using the weights of two unknown metal samples in air and their weight in water. These two things are the only data needed for the determination of specific gravity because of the efficiency brought about by deriving the formula. Table 1. Determination of Specific Gravity of Unknown Liquids Using Hydrometer TABLE 1.1. Determination of Specific Gravity of Unknown Liquids Using Hydrometer

Specific Gravity Name of Sample Percent Error

Sample 1

Sample 2

0.84

1.00

Alcohol

0.82

2.44%

4

Water

1.00

0.00%

The result for the alcohol that yields with a 2.44 % error can be a result of impurities and/or as a result of the water being tested first with the hydrometer and used when testing the alcohol without having dried completely. And with this data obtained , succeeding parts of the experiment can be done, particularly Procedure A for the fact that it must be the water to be used. Table 2. Determination of Specific Gravity of Unknown Solid Samples Heavier than Water TABLE 1.2. Determination of Specific Gravity of Unknown Solid Samples Heavier than Water Sample 1

Sample 2

Weight in air, 𝑊𝑎

19.90 g

44.90 g

Weight in water, 𝑊𝑤

17.60 g

27.20 g

Specific Gravity 𝑊𝑎 𝑆𝐺 = 𝑊𝑎 − 𝑊𝑤

8.65

2.54

Name of Sample Percent Error

Copper

8.89

2.68%

Aluminum

2.70

6.05%

Observing the data gathered in Table 2, it shows that in the two unknown liquid samples, the weight of the sample metal in air is greater than the weight of the sample metal in water. The reason for this is that because of the upward buoyant force, water exerts an upward force, which is the buoyant force, making the tension due to weight of the sample metal smaller. Having the specific gravity of 0.84 and 1.00, respectively, determination of the name of the unknown liquids will be easy. The unknown liquids are alcohol and water, respectively. Additionally, it can be seen that the loss of weight in liquid is lesser in alcohol than in water. Although it is not obvious that it is equal to the buoyant force of the liquid. Moreover, the trend goes that when loss of weight in liquid increases, then specific gravity also increases. So when the liquid is more buoyant, then the liquid has higher density. It has a greater force to rise up the object immersed on it. Furthermore, brass which is less dense than water has a displaced mass lesser than water. Moving on, the third part is the determination of specific gravity of unknown liquids using hydrometer. The percent error calculated was 0 % so the specific gravity gathered was accurate. For materials lighter than water, it is difficult to determine its specific gravity using Archimedes’ principle since the object will just float in water. In order to do this, a sinker was used.

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Table 3. Determination of Specific Gravity of Unknown Liquids TABLE 1.3. Determination of Specific Gravity of Unknown Liquids Sample 1

Sample 2

Weight in air, 𝑊𝑎

19.90 g

Weight in water, 𝑊𝑤

17.60 g

Weight in the liquid, 𝑊𝑙

18.00 g

17.70 g

Loss of weight in Liquid, 𝑊𝑎 − 𝑊𝑙

1.90 g

2.20 g

0.83

0.96

𝑊 −𝑊

Specific Gravity, Gravity 𝑆𝐺 = 𝑊𝑎−𝑊 𝑙 𝑎

𝑤

Name of Sample

Alcohol

Percent Difference

0.82

Water

0.74%

1.00

4.44%

Table 4. Determination of Specific Gravity of Solid Lighter than Water TABLE 1.4. Determination of Specific Gravity of Solid Lighter than Water Name of Sample: CORK Weight of cork in air, 𝑊𝑎

3.80 g

Weight of cork in air and sinker in water, 𝑊𝑐𝑎−𝑠𝑤

21.40 g

Weight of both sinker and cork in water, 𝑊𝑐𝑤+𝑠𝑤

17.80 g

𝑊𝑎 𝑐𝑎−𝑠𝑤 −𝑊𝑐𝑤+𝑠𝑤

Specific Gravity, 𝑆𝐺 = 𝑊

1.06

The overall volume displaced by the cork and the sinker will be the volume of the two components. Since mass and density of the sinker is known, we could easily substitute the value for the determination of the density or specific gravity of the unknown. When the weight of the cork in air, the weight of sinker alone and with the cork at water, we can compute for the specific gravity of the cork. The loss of weight of cork is simply the buoyant force exerted by the water to the cork.

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Conclusions Archimedes’ principle states that a body, when it is completely or partially immersed in a fluid experiences a buoyant force, which is equal to the weight of the fluid it displaces. This principle is a law that can be used to explain up thrust or buoyancy. Buoyant force is an up thrust or upward force exerted by a fluid on an object immersed init resulting in the apparent loss of weight of the object. In this experiment, we determined the density and specific gravity of solids and liquids following Archimedes’ principle. Density and specific gravity of materials are unique on each object that makes it as a tool in the identification of the material. Density is the ratio of the mass per unit volume. While, specific gravity is the ratio of the density of the material with the density of the reference liquid which is water. When an object is submerged in liquid, there is a buoyant force present in water pushing up the object. This buoyant force causes the object to lessen its weight. Furthermore, the buoyant force is also the weight of the liquid displaced by the object. The weight loss of liquid is in equivalent magnitude to buoyant force. We could derive for the formula on determining the specific gravity from buoyancy and the net force of the system. By that method, the density of the object can be readily determined. That’s why, specific gravity would be the measure of the relative density of the object compared to water. Considering Archimedes’ principle in real world, it helps to function the submarines, hot air balloons, ships and the like.

References [1] Halliday, Fundamentals of Physics, 9th edition. [2] http://www.brightstorm.com/science/physics/oscillatory-motion/archimedes-principle [3] http://www.physics247.com/physics-tutorial/archimedes-principle.shtml

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