1 Archimedes Principle - PDF

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Lab 1...

Description

Archimedes’ Principle and the Buoyant Force (rev. 1.1) Name

Objective:

To calculate and measure the buoyant force acting on either a submerged or floating object.

Apparatus:

Water, isopropyl alcohol (IPA), two 250-ml beakers, one 600-mL beaker, 1000-mL side-arm flask, short piece of rubber tubing, 100-mL graduated cylinder, large paper clip, PVC cylinder, paper cup, pennies, DataStudio, force sensor, Vernier calipers, digital scale, metal and wood blocks, “special tool,” 12” ruler, funnel.

Background: When an object is placed in a liquid, it often floats on the surface of that liquid. This occurs because the liquid exerts an upward force, a buoyant force, on the lower surface of the object that is greater than the gravitational force acting downward on the upper surface. Opposite and equal lateral forces will cancel out. The fact that a buoyant force always exists does not guarantee that the object will float; that depends on whether the object’s weight exceeds the maximum buoyant force. We will use Archimedes’ Principle to compute the maximum buoyant force (Fb) that a given liquid can exert on an object and predict if the object will float or sink. According to Archimedes’ Principle, the buoyant force, Fb, is just equal to the weight (WL) of the liquid it displaces when an object is placed in that same liquid.

Fb = WL

(1)

WL can be expressed as the product of the liquid’s density (ρL), g, and the displaced volume. (2) WL = g*(ρL)* Vdisplaced Combining equations (1) and (2) results in:

Fb = g*(ρL)* Vdisplaced

(3)

As the object is lowered into a liquid, the volume of liquid displaced, Vdisplaced, and thus Fb, increases. The largest value Vdisplaced can ever reach is the entire volume of the object itself (Vobject); this sets a limit on the largest possible value of Fb.

Fb(max) = g*(ρL)* Vobject

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If the object’s weight (Wobject) is less than Fb(max), the object will stop submerging before it completely sinks; hence, it floats. Wobject < Fb(max) Î it floats Wobject > Fb(max)

Î it sinks

The apparent weight (Wapparent) of an object in a liquid will be its true weight (Wtrue) minus the buoyant force the liquid exerts on it. This is the case whether the object is floating or not. Wapparent = Wobject - Fb (5) The object you will be using in this lab is a cylinder. Its volume is given by:

Vcylinder = π r2 h

(r=radius, h=cylinder length)

By knowing the object’s volume and weight, as well as the liquid’s density, you can determine if an object will float or sink.

Procedure: Part A. Determine the density of water and IPA Pour out a known volume (using the graduated cylinder) of each liquid and measure the mass of each with the digital scale. 1. Divide the mass by the volume to get ρL.(use units of g and cm3 = cc = mL) Parameter Volume (cm3 ) V

Water

IPA

Mass (grams) m Density (g/cm3) ρ Part B. Measure the buoyant force in water and IPA 1. Use the Vernier calipers and ruler to measure the PVC cylinder volume. hcylinder = _____ cm, dcylinder = _____ cm, X-Areacylinder = _____ cm2 Vcylinder = _________ cm3 2. Use the digital scale to measure the cylinder’s mass mcylinder=________g 3. Calculate the density of the cylinder ρcylinder=__________ g/cm3

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4. Based on density alone, predict if the cylinder will float in either liquid:

Floats in water (y/n?)______ Floats in IPA(y/n?)________ 5. Attach a short piece of clear tubing to the side-arm flask. 6. Record the mass of an empty 250-mL beaker. mbeaker = _________ g 7. Direct the tubing into a 250-mL beaker, there should be no “dips” along its length. 8. Fill the flask with enough water so some pours out into the beaker. 9. Discard this overflow water collected in the beaker. 10. Connect the force sensor to the DataStudio and use “digits” as the display. Click run. Adjust the size of the display to maintain 2 significant digits. 11. Attach a large paper clip to the force sensor hook. Zero the force sensor (while keeping it vertical) by pressing the “tare” button on the sensor’s side. 12. Now attach the cylinder to the paper clip; you should get a weight of about 0.75 N on the screen. Record the exact value in the following table. This is Wtrue. 13. Now slowly lower the cylinder into the liquid by hand. Be careful to collect all the displaced liquid as it flows through the tubing into the beaker. 14. Note the Wapparent reading as the cylinder is lowered. It will be changing. 15. The cylinder may float or sink − if it floats Wapparent will be zero. If it sinks there will be some non-zero apparent weight. Record your values at the point where the cylinder is clearly floating or has sunk (but not touching the bottom). 16. The measured buoyant force is: Fb = (Wtrue. - Wapparent). 17. Record the weight of the displaced liquid. 18. Repeat the procedure with IPA as the liquid. 19. DO NOT DISCARD THE IPA, POUR IT BACK INTO THE BOTTLE. True cylinder weight ,Wtrue in Newtons =________

Cylinder volume = __________cm3

Water

IPA

Apparent cylinder weight in liquid, Wapp. (N) Buoyant force Fb = Wtrue - Wapp

(N)

Displaced liquid weight, WDL

(N)

% difference between measured Fb and WDL (%)

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Part C. Effect of floating and submerged objects on the weight of a liquid filled beaker 1. Fill a 250-mL beaker about ¾ full with water. 2. Determine the weight individually of the metal and wood blocks. 3. You will be floating (or sinking) the wood and metal blocks in the water. To facilitate this, bend a paper clip into a hook/skewer (see drawing). 4. First, have a group discussion as to how these blocks will affect the change in weight (ΔW) of the (beaker + water) as measured on the scale, for the five situations below. Then calculate the expected changes in the table below. 5. Now, measure the beaker’s CHANGE in weight (ΔW) when the block is supported as shown below. FIRST: ZERO THE SCALE WITH THE BEAKER+WATER ON IT Special tool (for “pushing” on the water)

Volume of: either block ≈30. mL=30.cc special tool plate≈1 mL=1 cc

Use kg for masses below scale =_____ A. Special =_____ tool “pushing” on the water

mmetal. blk =_____ Wmetal. blk mwood blk . =_____ Wwood blk m30 mL H20=_____ W30 mL H20=_____

Paper clip Wood block

metal block

scale B. Metal block, resting on the bottom

scale C. Metal block, suspended by a hook (submerged)

scale D. Wood block, floating on the surface

scale E. Wood block, forced under water

ΔWeight, calculated (N) ΔWeight, actually measured (N) In the space below, write out the equation you used for each calculation (A-E) and briefly describe the reasoning behind the CHANGE in weight for each situation: A.

B.

C.

D.

E.

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Part D. Sinking a paper cup First, calculate and then confirm the number of pennies required to sink a small paper cup in water: 1.

Determine the weight of a penny in Newtons by measuring at least 10 as a group and dividing by that number. Penny weight = __________ Newtons You will need to compute the volume of the cup in cubic centimeters based on its dimensions. Your answer should be between 80 and 110 cc. The cup’s shape is called a frustrum and has the following formula for its volume:

Volume =

πh 3

( a + ab + b ) 2

2

a = _____ cm b = _____ cm h = _____ cm Volume cup = ______________ cm3 2.

Write out the equation you will use to calculate Fb(max) when the cup is floating in water. Fb(max)= ______________

3.

Now calculate: Fb(max) =

Newtons (in water).

4.

Determine how many pennies it would take to reach Fb(max): #P = Fb(max) / (1 penny’s weight) = _____________ This should equal the number required to sink the cup.

5.

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Add about half this number of pennies to the cup and then lower it carefully into the 600-mL beaker. Now, gently add additional pennies until it sinks the cup. Don’t drop them (remember about impact forces), and try to keep the pennies centered in the cup. Just before it sinks, observe the surface of the water at the cup’s lip. This is called “surface tension” and will have some effect on your data.

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Liquid = water

To sink the cup

# pennies (estimate) # pennies (actual) % difference

Questions: 1. (Part B) As the cylinder was being lowered into the water, how can you explain the continuously changing apparent weight?

2. (Part B) For the case of the floating cylinder, what caused its apparent weight to go to zero?

3. Since the cylinder sinks in IPA but floats in water, the buoyant force from the water must be greater. Explain why this is true even though the displaced volume in water is less.

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4. (Part C, test C). Describe the physical mechanism of how the submerged metal block increased the measured weight of the beaker without any solid contact to it. (Newton’s 3rd law is not a mechanism). To help answer this question, perform the following exercise: watch what the water in a beaker does when a block is slowly submerged into it.

5. (Part C). Explain why the change in measured weight of the beaker was nearly equal when either the metal or wood block was completely submerged (but not touching the bottom of the beaker).

6. (Part C). Both the Special Tool, and the block have the same “face” area of 30cm2. However, inserting the block into the water caused the beaker’s apparent weight to noticeably change while the Special Tool had almost no effect. Explain why inserting these two objects (with the same face area) produces such different results on apparent weight.

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7. For the cup-sinking operation, list what you think are the 4 most significant sources of experimental error in estimating the # of pennies (be specific). In each case, indicate whether the error will increase or decrease (↑ or ↓) the actual penny count.

8. Consider a distant planet where the gravity is much stronger than on earth. If you just barely float in water on earth, how would your buoyancy be affected in water on the strong-gravity planet (where your weight is greater)? Explain why in terms of concepts used in this lab.

9. Explain why a ship made out of steel (steel is ~8 times denser than water) can float in water. Use Archimedes’ Principle as the basis of your explanation.

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Pre-lab Quiz

Name

1. Do a little on-line research to determine the expected density of water and Isopropyl-alcohol (IPA) in units of g/cm3.

2. What is the reason that the apparent weight is less than the true weight of an object when it is submerged in a liquid?

3. Go online for some background and then give a brief explanation of the term “Surface Tension.”

4. Calculate the volume of a penny based on its physical dimensions (measure one with a ruler).

a. Penny thickness = ________ cm b. Penny diameter = _________ cm c. Penny volume = __________ cm3 (show work)

5. Calculate the expected mass and weight of a penny based on the volume and its density. (hint: assume it is a post 1982 penny, which is composed primarily of zinc.)

a. Density of zinc = __________ g/cm3 b. Penny mass = ____________ g

c. Penny weight = ____________ Newtons (show work)

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