Week 5 Group Problem

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131

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Physics

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Apr 3, 2024

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Physics 131 Group Problems: Week 5 Learning Goals . After working through this worksheet, students will be able to: 1. Use 1 the volume of fluid an object displaces to find the magnitude of the buoyant force. 2. Use the buoyant force to determine: a. Whether an object of known density will sink or float. b. The density of an object, given its acceleration. 3. Calculate the acceleration of an object due to its buoyant force and weight. 4. Use the Reynold's number to determine the type of drag force on an object in a fluid. 5. Use the drag force to find the terminal speed of an object in a fluid. Problem 1: Floating and sinking A piece of cork floats on the surface of water but a rock of the same size will sink. A. What can you conclude about the density of cork (pc) in relation to the density of water (pw )? (I.e. are the densities the same, is one greater than another, is there not enough information to tell, etc.) 1~ ~' ½ Si~ e,or't- lS \t~~ j,"'~ 4\1-c, ~Si~ ~"-<- \f.),-M./ B. What can you conclude about the density of rock (pr) in relation to the density of water (pw)? 1\-\t. WS i 1 +1,--.t ro t,,\C. l~ re M~ · \.-"'- ~ ~i fJt e&.rt i t.,A ~ r For both · the cork and the rock, there are at least two forces acting on the object in the vertical direction: the weight of the object (pointing straight down) and a buoyant force (pointing straight up). C. What can you conclude about the magnitude of these two forces on the cork while the cork is floating stationary at the surface of the water? (I.e. are the forces the same magnitude, is one force larger than another, is one of the forces zero, et.) "ilrt_ 'o u,'l °"'~ ~cwl(, i S ·~f"', t r- ~V\t ~-<w,,1-tl\}l ,) ~, t.,C, - ,,. 1 \l:c,, \.; lfutl D. What can you conc!ude about the magnitude of these two forces on the rock while it is sitting under the water, on the bottom of the container? W~ i+ \1 -lk- ~I -+W, tv~V~~I\ V\ / S lovo'1 ~+ ~r'-C.-

Problem 2: Buoyancy . . · 11 f; I . .all ) bmerged in a fluid W1 ee a An object with a volume Vo which is (wholly or parti Y s~ 1 Ir the fluid downward. buoyant force, provided there is gravity or P:rhaps something se) p~fi:~!" the volume of The buoyant force arises because the fluid bemg pulled down tnes to th fl .d water that is displaced by the object, by pushing the object upwards out of e U1 • · The magnitude of the buoyant force F 8 from the fluid(/) on the object (o) is = Pt vdn · · hi h · "" th way" of the · · Where Pt is the fluid's density, and Vd is the volume of the obJect w c 1s m e fluid (called the displaced volume). A. Assume your object is a box with volume Vo, and the fluid is water. 1. Come up with a physical situation in which the displaced volume Vd is less than the object's volume Vo (that is, describe something you could do to the box to make this happen). 2. Come up with a physical situation in which the displaced volume Vd is equal to the object's vqlume Yi,. ., , wv 5» • ~, -\({ ~ fr ' ~\l ~U \r ~ ½ ~ ', B. Draw a free-b?dy diagram for a box floating at rest on the surface of a lake (ignore any effect of the air). How do the magnitude of the forces in your diagram compare to each other? · · 2

C. The force of gravity has magnitude = m 0 g: Of course, for an object of mass mo, density Po and volume t{,, we know that Po = ~: · Solve for I in terms of Po, Va, and B (but without m 0 ). -~ - c q fo = _2.:: \ rf:" 0 \ ~~ 0 _ '{,;0 " \J 0 D. From your FBD you should see that the magnitudes of the buoyant force and gravity are equal. Use this to write an equation relating your expression for I above and the · ; : · definition of the buoyant force from the previous page, in terms of the given variables. E. A rock is more dense than water. However, as we have seen, if an object is floating, it will satisfy the equation you obtained in part D. Why can't a rock satisfy that equation? \Jo ~l ".._ vJ' \ \ 'lot- ""' \ ~o==\ltl\. F. The figure to the right shows three objects floating in water. 1. Rank, in order from largest to smallest, the densities of the objects A, B, and C. A'> c'> I 2. Explain how you use your ~quation from part D to determine your ranking. U1N\.~~-~ ~o W'l ~" 1 U1 f\J \~f\j ~ \, ;i \J11 \,-\ ~ \\. ,~ *~ -./', . G. The volume of water displaced by A is ¾ of its total volume; by Bis 2/5 of its total volume, and by C is ½ of its total volume. Use the density of water (Pr = 1 .... L) to . cm 3 calculate the densities of each of the objects using your equation from part D: PA= I 1r. I < \t• '·: ' - - - - j PB= 3

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Problem 3: Drag forces and sedimentation. In the previous problems we focused on objects at rest (~or which the buoyant force equaled the weight force). Next, we will analyze the motion of an object that sinks in a fluid - this could be a rock through air or a red blood cell through blood. - A. In the space below, draw and l~bel a free-body diagram for the object. B. Use the following questions to check that your FBD is correct. I. Since the object is sinking down through the fluid, is the magnitude of the buoyant force greater than, less than, or equal to that of the weight force? Make sure your FBD above reflects your answer. -- -> I F!>l L \fc-, \ 2. Since the object is now moving through a fluid, what direction does the drag force point? 1><~~ ~<-\. "C,\\,\-\1 \ ~t.) Drag forces increase as the object speeds up, so an object falling through a fluid will go faster · and faster until the drag is big enough that the net force is zero (this fastest speed is called the "terminal speed"). Later in lab you will see this illustrated with coffee filters in air and beads in liquid. C. Use Newton's 2 nd law to write an expression relating the forces on your FBD, when the object is at terminal speed. Co"5-%"--k- 'f<--ed- : r(lt.,+ -.: Ni~--() ~ f i - f 6i + 9 -:.. 1 '(D,, D. Consider a rock of mass 3.5 kg and volume 0.002 m 3 falling through air (with density kg 1. 2 m3). 1. What is the magnitude of the drag force once the rock is at terminal speed? I L ,_- \, (I f !i 1-- F ... - Q r Fci::o fv .. ~c-.-Fe, -= ""~ - r~~, 4

r ' - I ,, \ :. _- ~-- . \ ri 1) t ~~ •s it fl) 2 · In this case the drag force is inertial, so I.Fdragl = lftv I = Cvp 1 Av 2 • Assuming the rock has a drag coefficient Cv = 0.6, and cross-sectional area A = 0.02 m 2 , what is the r-0ck' s terminal speed in ~? , s 'L ,z,,_ fo IL cn~ . Lr-'f r A\/ :. Yc 1.. '. "' _ 'l <. . ' y \ ' - ' V t" ' ,. . " . ,. · , f:> . ' I ! . '.' · ' , . \ • · - C...t5q." ,A ., 3. 1 is about 2.2 mph. Does your answer above seem reasonable? If not, check s your math again. E. Consider a red blood cell of mass 45 pg (1 pg = 10- 15 kg) and volume 40 µ m 3 falling through blood plasma (density 1025 !~). "" "'~s :::- 4 .6 d(J -;'!> 1 to luN-v : lf .,1 00 - ' 1 "" 3 1. What is the magnitude of the drag force once the / cell is at terminal speed? (Hint: It might be easier to first convert all the units first, and then start calculat~.) I - ,,;. \::- 1 '!. '1 0 l \ i L-/5~~• 0 j_ ;: '1, ~•lt," I ytv' • l,1t, · '~1"'""~:1.j,{O"''I • I f' J 1i t[ £,+fD: Fci ~p = F J - f" .: '1 . I l'.J - l4 N 2. In this case . the drag force is viscous, so IFdragl = IFvl = 6m,rv. If the cell has a radius r = 3.5 μm and the viscosity of blood is T/ = 0.01 Pa · s = 0.01 N·;, what m is the cell's terminal speed? t. i I ( ) . (' I . I ' : () -· ,,, ;; .. (, \.\ \' - I 'I 3. At this terminal speed, how many days would it take the cell to fall 5 cm (about the length of a test tube)? Your result in part E above illustrates one of the reasons why we need a centrifuge to sediment cells to the bottom of a test-tube. Spinning in a centrifuge increases the apparent weight of the cells, as we will learn.next week. · 5

Problem 4: Life in the slow lane. Microorganisms like bacteria have evolved to develop flagella that help propel them in water. The figure on the right shows a simple model of a flagellum that uses rotary motion (like a rotating cork-screw) to apply a continuous push to the fluid. From Newton's third law, the fluid pushes back on the bacterium. This Cell membrane"" ' . i~ (equal and opposite) force propels the bacterium \ forward. Once the bacterium starts to move, a drag ProJ?eller-like force from the surrounding fluid opposes its motion. motion The drag force increases as the speed increases, until the drag force is equal in magnitude to the propelling force, at which point the bacterium moves at terminal speed. A. In the space below, draw a FBD of the bacterium swimming in water at a constant velocity. Pick the direction of the velocity in the positive x-direction. Include both horizontal and vertical forces. .... L "" · j The drag force can be either inertial (I.FD I oc v 2 ) or viscous (I.Fv I oc v), and in most cases the other drag force is negligible. To determine which type is applicable in a given situation, we need to compute the Reynolds number, Re= PtLv_ Re> 2000 indicates that the inertial drag 71 force dominates, while Re < 1 indicates that the viscous force dominates. (In between both are significant, and the behavior is complicated.) B. Assume the bacterium is a sphere (with the "length scale" L = diameter= 1 µ m), swimming in water with Pt = 1000 k~ and T/ = 0.01 Pa · s, and a terminal speed of m . . 30μm. s 1. Compute the Reynolds number in this case. The Reynolds number should be dimensionless, so make sure that you convert units appropriately so that they all ' cancel out. 1 ooo , I i,1,.., • 'i> o _.":\ ' l'l. ,.. 0 ...... r _-:,.. .J C,v o.o\ · r ... . , 1 • 2. Which drag force is important in the life of _ a bacterium: the inertial drag force or the viscous drag force? ,~ i~tr~"-\ ~('j 6

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C. For a spherical object ofradius R, IFvl = · frrrTJRV. Find the magnitude of the drag force on the bacterium, in pN (recall ! pN = 1 x 10- 12 N). L=- 1 V (<.. == ';_ .:: 0 · \' """' . \r~ I How much force do the flagella of the bacterium need above velocity of 30 µ m/s? s~ ~s \f .J \ E. What is the weight of the bacterium, assuming its density is 1200 k~ ( 0% higher than m that of water)? F. Compare the weight of the bacterium to the magnitudes of the drag force and the force exerted by the flagella. Which of the forces on your free-body diagram are significant? fo 7

Week 5 Group Problem

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