Physics Lab 2: Archimedes Principle

Transcription

1 -1 - Physics Lab 2: Archimedes Principle , James J. DeHaven, Ph.D. The legend ges that Archimedes was asked by the King f Syracuse (in Sicily, nt New Yrk) t figure ut whether the crwn he wre was made f pure gld r nt. Since he was unable t find the vlume, he did nt knw hw t ascertain the density. The answer, t weigh the crwn in water, is said t have ccurred t him in the public baths, frm which, in his jy, he exited shuting, Eureka!. (I ve gt it!) In the prcess f articulating the prblem, and thinking thrugh its slutin, Archimedes stumbled upn the principle which bears his name tday: If an bject is submerged in a fluid, the fluid supprts the bject with a frce equal t the weight f fluid displaced by the bject. We call the frce that supprts an bject the buyant frce n the bject. If the weight f a submerged bject is greater than the buyant frce, the bject sinks. If the weight f a submerged bject is less than the buyant frce, the bject will rise t the surface f the fluid. If the weight f a submerged bject is exactly equal t the buyant frce, the bject will hang mtinless in the fluid. A flating bject submerges a fractin f its vlume s that the buyant frce exactly equals the weight f the bject. In tday s lab yu will explre the physics f fluids in the fllwing fur areas: 1) A direct cnfirmatin f Archimedes Principle. 2) Determinatin f the density f a metal. 3) Determinatin f the density f a substance less dense than water. 4) Determinatin f the density f a liquid Methd In rder t cmplete tday s lab, yu will need tw capabilities: 1) Yu will need t be able t weigh bjects under water; 2) Yu will need t be able t accurately determine vlumes. Vlume will be btained by measuring the water displaced by the test bject in a graduated cylinder. Weighing will take place n a cnventinal triple-beam r electrnic balance which has been utfitted t weigh hanging masses. If yu are using a mechanical balance, the fluid will be placed in a beaker n the swing-ut supprt (see figure 1). The mass t be weighed will be suspended frm the lwer hk. The balance pan will hang freely frm the upper hk as shwn in figure 1.

2 -2 - Balance Beam Metal Sample Supprt Arm Balance Pan Swings Freely Fig. 1 Mechanical balance setup fr weighing an bject submerged in a fluid. If yu are using an electrnic balance, the fluid will be placed in a beaker n the plastic beaker supprt, with the balance pan remved (see figure 2). The mass t be weighed will be suspended frm a wire hk.

3 -3 - Balance Hk String metal sample Supprt Stand Electrnic Balance Fig. 2 Electrnic balance setup fr weighing an bject submerged in a fluid. Nte that the balance pan is remved and replaced with the beaker supprt stand Experimental A. Experimental Verificatin f Archimedes Principle 1. Measure the mass f the cylindrical brass sample 2. Measure the vlume f the cylindrical brass sample. Yu will need t use an electrnic balance t accmplish this. Use a graduated cylinder t measure the vlume. Carefully place the sample in a cylinder and add water until it is cmpletely submerged. With an eyedrpper, add water until the level exactly reaches sme easy-t-read reference value. Remve the weight and shake any water drps back int the cylinder. Weigh the cylinder, fill back up t the reference mark, and weigh again. The difference between the tw masses can be used t deduce the mass f the water which has been displaced by the hanging weight. Since the density f water is 1 g/ml, the number f ml f water displaced equals the difference between the tw weights.

4 -4-3. Nw weigh the cylindrical brass sample while it is suspended in water using the setup shwn in Fig. 1 r Fig. 2 (depending n yur balance). 4. Accrding t Archimedes Principle, the buyant frce shuld equal the weight f the water displaced by the bject. T understand the idea f the buyant frce, cnsider the fllwing free bdy diagram fr an bject submerged in sme fluid: F B (Buyant Frce) Figure 3: Free Bdy Diagram fr a Submerged Object m g (Weight) The buyant frce, FB, is the frce that the fluid exerts n the bject in rder and that pushes the bject up. Whether the bject will sink r rise t the surface will depend n the directin f the NET FORCE. We write an expressin fr the net frce, by examining figure 2 abve: [1] F net m g F B The behavir f the bject will depend n whether the net frce is greater than, less than, r equal t zer: If 0 F net [2] If > 0 F net If < 0 F net Object neither rises nr sinks Object sinks Object rises t surface

5 In this experiment, yu weigh an bject that is suspended under water. What yu are really measuring when yu weight this bject is the net frce in equatin 1. We call this weight THE APPARENT WEIGHT, w : [3] w' m' g Actually, when yu use a balance yu get yur result in terms f m, the APPARENT MASS f the bject. This desn t mean that the bject has a new, different mass when it is under water; mass is mass. It means that it feels r appears t have a smaller mass, and we call this smaller mass the apparent mass. Since the net frce is equal t the apparent weight, we can rewrite equatin [1] as fllws: -5 - [4] w' m g F B But the buyant frce is just equal t the weight f the fluid displaced (wf ) as shwn in equatin [5]. [5] F B w f Substituting [5] int [4] gives [6] [6] w' m g w f Then use weight mg t get [7]. [7] m' g m g m g f [8] m' m m f Cancel ut g t get [8] and nw substitute the definitin f density t get [9]. [9] m' ρ V Because m ρv [10] V But we knw that the vlume f the bject is the same as the vlume f the fluid that the bject displaces. Equatin 10 simply states this. Substituting [10] int [9] gives an expressin which predicts the reading we shuld get n the balance accrding t Archimedes Principle: [11] m' ( ) V ρ Cmpare yur measured value fr the apparent mass t that predicted by Archimedes principle.

6 B. Measurement f Densities Using Archimedes Principle 1. Yu will use the density f the brass cylinder which yu cmputed frm the data yu tk in part A Using Archimedes Principle determine the density f a) the aluminum cylinder, b) the plastic cylinder (see special directins belw) c)the irregular metal bject using Archimedes principle. D nt directly measure their vlumes. Instead submerge the bjects under water (nte: use the special directins belw fr the plastic bject) and use the weight thereby btained t determine the densities. Recall that the actual mass f the bject, m, is given by: [12] m ρ V while the mass that we measure while the bject is submerged, m, the apparent mass, is given by equatin [9]: [9] m' ρ V The difference between these tw masses can be btained simply by subtracting equatin [9] frm equatin [12]: [13] m m' Use [13] and [9] t relate the rati f the masses t the rati f the densities as fllws: [14] m m' m ρ V We then slve [14] fr ρ, the density f the submerged bject: [15] ρ m m m' ( )

7 Thus, since we knw the density f the fluid (water) the densities f each f the three materials can be determined. In each case, yu need m (the mass f the bject), and m (the apparent mass f the bject when it is weighed under water) fr the sample Special Case f Plastic In the case f the plastic bject yu will have t weigh it dwn t keep it under water. This means that yu can t directly measure its apparent mass--yu have t deduce it frm things that yu can measure. Use the brass weight (a rectangular slid mad f brass is prvided fr this purpse) t weigh dwn the plastic cylinder: Tgether, the brass weight and plastic cylinder will have a cmbined apparent mass, m', when weighed under water. Tie the brass and plastic weights tgether and then suspend the assembly and weigh under water. Brass Weight Plastic Cylinder Figure 4: Brass weight and plastic cylinder tight tgether with a string The measured apparent mass, m, is just the sum f the cntributins frm the brass weight and frm the plastic cylinder: [16] m' m' + BW m' P

8 -8 - The prblem here is that we dn t knw m p, the apparent mass f the plastic. Hwever, we can easily find m BW, simply by weighing the brass weight under water separately. We knw m frm measuring the apparent mass f the brass weight and plastic tgether, s it is easy t slve fr the apparent mass f the plastic: [17] m' P m' m' BW When yu d this calculatin, m p will cme ut negative. This is expected and des nt mean anything spky is happening, like anti-gravity. What is the reasn that yu expect this behavir? Once yu have fund the apparent mass f the plastic, yu can use an expressin analgus t equatin [15] t find the density: [18] ρ P P m ( ) P m m' P C. Measurement f Density f a Fluid Using Archimedes Principle Use the brass blck t determine the density f Wessn Oil, and predict whether the il will flat n water r sink t the bttm. Since yu already knw the density f the brass blck, by using the m yu already measured and by measuring m in Wessn Oil, yu can rearrange equatin [15] t calculate the density f the new fluid, Wessn Oil. Will Wessn Oil sink r flat n water? [19] ρ m m' f ρ( ) m

9 D. Please clean the beakers thrughly at the end f the lab. Equipment: On Tables eyedrpper r pipet Wessn Oil Paper Twels physics string (1 rll/table) Wash bttles with distilled water (2/table) In plastic bxes ml grad cylinder 200 ml tall frm beaker 400 ml beaker balances, pwer supply, black plastic cver, Archimedes wire supprt 5 masses: sq. brass, cyl. brass, cyl. Al, cyl. plastic, irregular metal Scissrs ruler Reprt: Intrductin: Write a brief intrductin cncisely stating the purpse f the experiment, and a cncise summary f the methds that will be used. Experimental: Describe the experimental apparatus and precisely what variables will be measured and hw they will be measured. Results: Summarize the results f the experiment. Shw sample calculatins. If yu are attaching cmputer generated tables r graphs, briefly explain them here. Discussin: Explain the significance f yur results and their cnnectin with mre general physical principles. Where it is pssible, cmpare yur numbers with accepted values. Explain any surces f errr.