Archimedes' Principle

INTRODUCTION

An object is submerged in water. The buoyant force on it is measured by weighing the water it displaces.

The buoyant force is also determined by measuring the difference between the object's weight in air and its apparent weight in water.

Some of the objects have the same density, some have the same volume, and some have the same mass.  The density of each object is measured and the dependence of the buoyant force on density, mass, and volume is explored.

Part I:  Finding the Buoyant Force Using Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object, completely or partially immersed in a fluid, is equal to the weight of the displaced fluid.

PROCEDURE

For each of the objects, find the weight of the water displaced by each one:

1. Find the mass of the beaker. Put the beaker under the overflow can spout as shown in left figure.

2. Pour water into the overflow can until it overflows into the beaker. Allow the water to stop overflowing on its own and empty the beaker into the sink and return it to its position under the overflow can spout without jarring the overflow can.

3. Tie a string onto each of the objects.

4. Gently lower the first object into the overflow can until it is completely submerged. Allow the water to stop overflowing. Find the mass of the water plus beaker. Subtract the mass of the beaker to determine the mass of the water alone. Optionally, multiply the mass by the acceleration due to gravity to find the weight of the displaced water.

5. Repeat this procedure for the other objects. Note that the plastic cylinder will float so don=t try to completely submerge it in the water. Also find the mass (or weight) of the displaced water when only half the brass cylinder is submerged.

6. List the objects in order from least buoyant force to greatest buoyant force. Is this in the same order as the mass list, the volume list, or the density list?  Are any of the buoyant forces nearly the same?  Why or why not?

Part II:  Finding the Buoyant Force by Finding the Upward Force

THEORY

When an object is submerged in a fluid, the apparent weight of the object is less than the weight in air because of the upward buoyant force (see Figure below). Thus the buoyant force can be calculated by finding the difference between the weight of the object in air and the apparent weight of the object when it is submerged in water.

PROCEDURE

1. Put the triple-beam balance on top of a stand as shown in Figure. Tie a string to the bottom of the pan and put a paperclip hook on the end of the string.  Zero the balance.

2. Hang the first object from the string. The balance will read the same as when the object is placed on top of the pan. Optional: multiply the mass by the acceleration due to gravity.

3. While the object is still hanging from the balance, submerge the object in a beaker of water so that the entire object is under water but it is not touching the sides or bottom of the beaker. Record the reading on the scale and, optionally, multiply by gravity to get the apparent weight. The scale reads in units of mass: Does the mass of the object change when it is submerged in the water?  What actually changes?

4. Calculate the buoyant force by taking the difference between the mass (or weight) in air and the mass (or weight) in water.

5. Repeat these steps for all the objects. Note that the plastic cylinder will float so do not try to completely submerge it in the water. Also, for the half-submerged brass cylinder, find the apparent mass (or weight) in the water when only half the cylinder is submerged. NOTE: The mass (or weight) in air of the brass cylinder is still the whole mass (weight).

6. Compare the buoyant forces found by this method to those found using Archimedes' Principle.

Method 1                                                                                       Method 2

In summary, find the buoyant force via two methods:

1) An object is submerged in water. The buoyant force B1 on it is measured by weighing the water it displaces.

B1 = ρfgVso = ρfgVdf = weight of displaced fluid      Archimedes principle

2) Buoyant force B is found from the difference between the object's weight in air and its apparent weight in water.

T2 + B = W => B = W - T2

 Mass of Beaker (MB) = 50.9 gr Object MB + Water (gr) MBW "B1" = Mass of Water (gr) MBW - MB Mass in Air (gr) MIA Mass in Water (gr) MIW "B" = (MIA - MIW) (gr) % Difference 100%|B1 - B|/B Al Cylinder (Silver) 73.4 73.4 - 50.9 = 22.5 60.6 39.7 60.6 - 39.7 = 20.9 8% Al Block Brass Cylinder (Gold) Brass Block Al Irregular Plastic Cylinder Brass Cylinder 1/2 sub.

 Mass of Beaker (MB) = 50.4 gr Object MB + Water (gr) MBW "B1" = Mass of Water (gr) MBW - MB Mass in Air (gr) MIA Mass in Water (gr) MIW "B" = (MIA - MIW) (gr) % Difference 100%|B1 - B|/B1 Al Cylinder (Silver) 73.2 73.2 - 50.4 = 22.8 60.4 39.8 60.4 - 39.8 = 20.6 10% Al Block Brass Cylinder (Gold) Brass Block Al Irregular Plastic Cylinder Brass Cylinder 1/2 sub.

Questions:

1. In each case, is the buoyant force that was determined using the upward force equal to the weight of the water displaced?

2. Which objects had the same buoyant force when submerged? Why?

3. For the plastic cylinder, what was the apparent weight in water?

4. How was the buoyant force for the totally submerged brass cylinder related to the buoyant force for the half-submerged brass cylinder?

5. What does the buoyant force depend on:  The mass of the object, or its volume, or its density, or the material from which it is made?