Density, Buoyancy and

Archimedes’ Principle

UM Physics Demo Lab 07/2013

Pre-Lab Question

What twovertical forces act on an object floating in equilibrium, partially submerged in a liquid?

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Property of LS&A Physics Department Demonstration Lab

Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan48109

EXPLORATION

Materials

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Property of LS&A Physics Department Demonstration Lab

Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan48109

1 bent fork

1 paper clip

1 Petri dish

1 brass block

1 aluminum block

1 wood block

1 Teflon rod

1 wood rod

1 aluminum rod

1 film canister

1 portion of fine sand

1 plastic spoon

1 digital gram scale

1 cut-off 2 liter soda container (vessel for floating film canister)

1 clear plastic ruler

1 calculator

2 paper coffee filters (sand containment)

Shared Components:

soapy water

paper towels

vacuum cleaner for sand cleanup

1

Property of LS&A Physics Department Demonstration Lab

Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan48109

Density

Densityis the ratio of mass to volume of a uniform material. You are going to measure and compare the densities of some common materials.

1. Objects float or sink in water depending on their density. Fresh water has density of about 1g/cm3. Predict which blocks and cylinders will float or sink in Dennison water using the cut-off 2 liter soda bottle container, then test your predictions.

Object / Float or Sink (Prediction) / Float or Sink (Observation)
Brass Cube
Aluminum Cube
Wood Cube
Wood Cylinder
Teflon Cylinder
Aluminum Cylinder
1. Measure the dimensions of the blocks using your ruler (use cm):

Brass / Aluminum / Wood
Length (l)
Width (w)
Height (h)
1. Measure the dimensions of the cylinders (use cm):

Dimension / Wood / Teflon / Aluminum
Height (h)
Diameter (d)
• Measure the mass of the blocks and cylinders in grams with the scale and record in the table below.
• Calculate the volume of the blocks and the cylinders. The volume of a block is:
• The volume of a cylinder is the area of the base times the height:
• Density (ρ, theGreek letter “rho”) is defined as the ratio of mass (m) to volume (V):
• Calculate the density of each block and cylinder from your mass and volume measurements and record your results in the table below:

Object / Mass
(grams) / Volume
(cm3) / Density
(g/ cm3)
Brass Cube
Aluminum Cube
Wood Cube
Wood Cylinder
Teflon Cylinder
Aluminum Cylinder
1. Which has a larger density: a 10 lb. bag of marshmallows or a 10 lb. bag of lead?
1. Which has the larger volume the marshmallows or the lead?
1. Based on your observations for questions 1 and 3, what relationship between the density of an object and the density of water determines whether the object floats or sinks in water?
1. Measure the density of water in Dennison. Use the graduated cylinder. First, measure the mass of the cylinder alone, and then measure the mass of the cylinder with some water in it (preferably an even volume such as 20ml). Subtract the mass of the cylinder from the mass of the cylinder with water in it to find the mass of the water. Milliliters are a unit of volume, 1 ml = 1 cubic centimeter. Calculate the density of Dennison water in g/cm3. Show your calculations!

(grams) / Mass of Graduated Cylinder and Water (grams) / Mass of Water Only (grams)

Density of Dennison Water: ______

Archimedes’ Principle

Archimedes’ Principle states that the upward buoyant force on anobject fully or partially immersed in a fluid is equal to the weight of the fluid it has displaced, which is in turn equal to the product of thesubmerged volume of the object, the density of the fluid and the acceleration of gravity. For the object to float in equilibriumthe upward buoyant force must equal the downward weight of the object, which is in turn the product of the object’s mass and the acceleration of gravityg.

TestingArchimedes’ Principle

Carefully measure the diameter of the bottom of the film can with your ruler and record in the table below. Next,place the film can inside the coffee filters and fill the can with 2-3 plastic spoonfuls of sand andtightly seal the lid. Now float the can bottom down in the 2-liter soda bottle, next to the wall of the bottle.If necessary, add or remove some sand until the can floats vertically with the lid clear of the water. Carefully measure the distance from the surface of the water to the bottom of the submerged can. Calculate the volume of the portion of the can which is submerged (see question 3) and calculate the corresponding mass of water the can has displaced(submerged volume x density of water). Finally, dry the can, weigh it with the scale and record the can’s mass below.

Diameter (cm) / Submerged Height (cm) / Submerged Volume (cm3) / Mass of Can
(g) / Mass of Displaced Water
(g)
1. Draw a Free Body Diagram for the film can floating in water. Indicate the surface of the water and include the buoyant force and the object’s weight. Express the magnitude of the buoyant force in terms of the mass of the displaced water and the weight of the can in terms of the can’s mass. How do the magnitudes of these two forces compare?
1. In light of the answer to question 8, what relationship must hold between the mass of the displaced water and the mass of the film can? Justify your result by writing a simple expression relating the magnitude of the buoyant force and the weight of the can consistent with both Archimedes’ Principle and the fact that the can is floating in equilibrium. (Hint: consider your answer to question 8).
1. Compare the mass of displaced water you have estimated for the floating film can and the mass of the can. In light of your answers to questions 8 and 9, do your results support the validity of Archimedes’ principle? Explain.

Surface Tension

1. Pour water into a Petri dish as full as possible. How far over the rim can you go before it overflows? Draw a picture of the shape.
1. How is it possible for the water to extendabove the rim of the dish without spilling?
1. Prediction: Paper-clips are made out of steel with a density of 7 g/cm3. Can a paper-clip float on water? Explain your reasoning.
1. Place a paper-clip in the water by carefully lowering it onto the water’ssurface with the bent fork. Does the result agree with your prediction? Explain your observations. Are these observations consistent with Archimedes’ Principle? Explain. (Hint:Sometimes the paper clip will perform better if it is somewhat greasy; consider rubbing it with your fingers or on your forehead or nose).

APPLICATION

1. While floating a paper clip, add a drop of soapy water with your finger onto the surface of the pure water. Place the drop on the far side of the Petri dish from the clip. Observe what happens. Discuss with your group what you’ve observed and record your conclusions about what happened.

Everyday Applications

1. Some redwood trees on the coast of Northern California grow to a height of 122 m (367 ft) and a diameter of 7 m (22 ft). The mass of one of these trees is 2.2 x 106 kg (4.6 million pounds!). If one of these massive trees fell into the water, would it float or sink? Explain your reasoning and show your calculations. (Data: In SI units the density of water is 1,000 kg/m3.)

Summary

• Density is the ratio of an object’s mass and volume
• Archimedes’ Principle states that the buoyant force on an object is equal to the weight of the fluid it has displaced.
• As a consequence of Archimedes’ Principle, an object will float if its density is less than that of the fluid, sink if its density is greater than that of the fluid and be neutrally buoyant if its density exactly equals the density of the fluid
• Water exhibits a surface tension which causes the surface of water to behave like a membrane. Small objects with a density greater than that of water can be supported on the water’s surface in violation of Archimedes’ Principle.
• Soaps and detergents dispelthe surface tension of water and eliminate water’s ability to support objects on its surface.
• The most famous use of a density measurement was done by Archimedes (290-210 B.C.). He was hired by the king to verify that a gold crown he’d commissioned had not been substituted in part by less valuable (and less dense) silver. However, the crown was very expensive and Archimedes could not melt the crown to ascertain its volume. While stepping into a bath he observed the change in water level, and realized he could do a volume displacement measurement. In his excitement, he ran naked through the streets yelling “Eureka! Eureka!” (Greek for “I have found it!”).
• Product manufacturers (in industries such a food, beauty, and health) use density measurements to verify consistency among products. They measure the density of product from different batches to make sure that there is consistency. Distillers of illegal “moonshine” liquor will often adulterate their product with ash or other substances to alter the density to mimic a higher alcohol content than the liquor actually contains.

Final Clean-up

Please clean all table surfaces you used and return equipment to the carts. Please dry individual pieces that may be wet and dispose of water in the sink. Please sweep any spilled sand into the trash.

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Property of LS&A Physics Department Demonstration Lab

Copyright 2006, The Regents of the University of Michigan, Ann Arbor, Michigan48109