Preparation
Prepare for this week's quiz by reviewing angular motion and moment of inertia. Read the sections of your textbook that cover Archimedes' Principle, buoyant forces, density, and specific gravity.
Principles
The density,
ρ
(pronounced "rho"), of an object is the ratio of its mass to its volume
ρ = m
V
.
Since very few substances have exactly the same density, this property can be used to identify an element or compound.
The density of water is 1 g/ cm3 which is the same as 1 x 103 kg/m3. The ratio of the density of any solid or liquid or to that of water is called its specific gravity ( S ). Since it is the ratio of two densities, specific gravity has no units and is independent of the system of measurement.
When you are swimming in a pool or in the ocean you seem to weigh less. This is because the water exerts a buoyant force on you. The apparent loss of weight that you feel when you are submerged in the water is equal to the buoyant force exerted you by the water. Salt water is denser than fresh water and exerts more force; your apparent weight is less in salt water than in fresh water. In some extremely salty bodies of water like the Salton Sea or the Dead Sea it is almost impossible to sink.
In other words, if you weigh an object in air and then weigh it while it is submerged in a liquid, its weight in the liquid, called its "apparent weight" will be the smaller of the two. This is because the liquid exerts a larger buoyant force on the object than air does.
One way to understand buoyant forces is to consider an element of volume,
ΔV
, somewhere in a container of liquid. This element is in equilibrium, that is, it is neither rising nor falling in the container. This means that the weight of the liquid, which acts downward, is exactly balanced by an upward force provided by the liquid below it. This upward force is the buoyant force. If the volume of liquid, ΔV , is replaced by an object of the same volume, the object will feel the same force. The magnitude of the buoyant force, therefore, is equal to the weight of the displaced liquid
F = mg = ρ gΔV b where ρ is the density of the liquid, ΔV is the volume of the submerged part of the object, and g is the acceleration due to gravity. The force is directed upward. This is called Archimedes' principle. He is said to have discovered this principle while he was taking a bath.
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If the object's weight is less than the weight of the displaced liquid, it will float. It may also be able to support additional weight. The amount of weight a floating object can support is the difference between the object's weight and the weight of the liquid that the object would displace if it were totally submerged. An object that floats will have negative apparent weight when it is totally submerged.
The specific gravity of a liquid can then be measured by finding the ratio of the buoyant force an object feels in the liquid to the buoyant force it feels in water. This is the same as the ratio of the apparent loss of weight in the liquid to the apparent loss of weight in water
S = w - w
1 w - w
2 where w , w , and
1 w
2
represent the object's weight in air, in the liquid, and in water, respectively. Since g is a common factor in the denominator and the numerator it can be factored out and the specific gravity can be written in terms of the object's apparent mass in air, liquid, and water
S = m - m
1 m - m
2
.
Equipment
1
1
1
1
1
1
1
1
3. pan balance graduated cylinder beaker of each kind of metal cylinder wooden block jar
Vernier caliper rubber band extra masses string
Procedure
Use the CGS system of units. Measure masses to 0.1 g. Measure lengths to 0.1 mm. Do not get water into the unknown liquid.
1. Find the mass of each cylinder. Find the mass of the wood block and the empty jar and lid together.
2. Use the Vernier caliper to measure the height and the diameter of one cylinder. Do the same for the jar. Measure the length, width, and height of the wood block.
Set a beaker of water on the platform attached to the balance. Tie a piece of string to a rubber band. You will use this to make a sling that you can use to lower the cylinders into the water so that they are completely submerged, but not touching the bottom. Be
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sure the pan is still hanging from the arm of the balance and tie the other end of the string to the hook that the pan hangs from. Make sure that the string does not touch the sides of the beaker. Find the apparent mass of each cylinder in water.
5. Your instructor will have a balance set up with a beaker of ethanol. Carefully dry your cylinders and find the apparent mass of each one in the ethanol.
6. Calculate the volume and the density of the wood block and enter this into your data section. From this, estimate how much of the wood will be above and below the level of the water when you put it into your beaker of water. Record your estimate.
7. Put the wood block into the beaker of water. Use your ruler to measure how much of the block is above and below the water level.
8. Calculate the volume of the plastic jar and enter this into your data section. Estimate how much mass you can put into the jar so that it will float but be completely submerged.
Record your estimate.
9. Put your estimated amount of mass into the jar, close it and drop it into the beaker.
Record what happens. If the jar sinks, or is not completely submerged, adjust the amount of mass until you get the effect you want. Record how much mass you had in the jar.
10. Take the masses out of the jar and completely fill it with water. Put the lid on it and find the mass.
11. Dry everything you used. Put all your equipment away and be sure your instructor signs your notebook
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Data
Your data should consist of the dimensions of the cylinder, the block, and the jar, the masses of the cylinders, the block, and the jar in air, the apparent mass of the cylinders in water, and the apparent mass of the cylinders in ethanol. It should also have your volume and density calculations for the block, your estimated and actual amount of wood above the water line, the volume of the jar, your estimated and actual values for the amount of mass the jar will support and the mass of the jar when it is full of water.
Analysis
Put as much of you results as possible in a table.
1. Calculate the volume of one of the cylinders. The volume will be the same for all the cylinders. Calculate the density of each. Find a table in your textbook or on the web to decide what common material each cylinder is made of. Remember, these are common materials. Find the percent error from the accepted value.
2. Calculate the specific density of ethanol and compare it to the accepted value of .79.
Questions
Correct answers to these questions will require a least a paragraph each.
1. Compare the loss of apparent mass for each of the metal cylinders. What general conclusion can you draw from your results? What do you think you can say about the relationship between the density of a material and the buoyant force it feels? (Hint:
Think about Archimedes’ Principle.)
2. How did you predicted and measured value for how much of the wood would float above water agree? What can you say about the density of an object and whether it floats or sinks? What can you say about the density of an object and how much of it sinks below the level of the water? How much additional mass do you think your wooden block could have supported without sinking? Explain your reasoning
3. How did your prediction agree with experiment for the plastic jar? How did the amount of mass you could put in the jar compare with the mass of the amount of water the jar held? Remember to take the mass of the jar into account. What general statement can you make from this part of the experiment?
4. Archimedes is said to have curried favor with a king by determining that a crown the king had commissioned was not pure gold. Explain in detail how he could have done this using only a small sample of pure gold.
If it applies to you, write " I have not cheated on this lab report " and sign your name.
Grading
4 pts Data and Analysis.
4pts For each question
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