My laboratory

My laboratory

Archimedes' Principle

Purpose: To verify Archimedes' principle, and to determine the density of a fluid.

Discussion: Archimedes' principle states that the upward buoyant force B acting on an object which is partially or completely submerged in a fluid is equal to the weight of fluid displaced:

B = rVg

where r is the density of the fluid, V is the volume of fluid displaced, and g is the gravitational constant.

An object that is floating at the surface of a liquid is in equilibrium. According to Newton's Second Law the net force on a floating object must be zero. The gravitational force on the object (weight, W) must be balanced by the buoyant force, B:

B - W = 0

B = Mg

where M is the mass of the object.

In this experiment we will use a partially submerged test tube . As additional mass is added to the test tube it will submerge further, but always maintaining the balance of buoyant force and weight. By measuring the mass M of the test tube and the volume of water displaced V by the test tube it is possible to verify Archimedes' Principle.

B = Mg

rVg = Mg

rV = M

Procedure:

1. Measure the outer diameter D of the test tube using vernier calipers. Calculate the cross sectional area in cm²: A = pD²/4.

2. Number the centimeter marks along the white edge of a piece of graph paper. These marks define the y-coordinate. Roll up the piece of graph paper, and insert it into the test tube so that the zero on the scale is at the bottom of the straight part of the tube.

3. Fill a large container full of water. Fill the test tube with just enough small weights (machine nuts) such that the test tube will float vertically in the water. Be careful that you do not get any water in the test tube. Read the submersion depth y to the nearest 0.1 cm.

4. Determine the volume V of water displaced by the tube from your values of y and A. Estimate the error in volume due to neglecting the contribution of the rounded end of the test tube.

5. Remove the tube from the water, wipe off any water on the outside of the tube, and measure the mass M of the tube in grams, including the paper and weights.

6. Determine the average mass value for the machine nuts (only use one size nut in your experiment!). Adding one nut at a time, determine

(a) the mass of the test tube and its contents, and

(b) the volume of the water displaced by the test tube.

Take as much data as possible without allowing the test tube either to sink or to rest on the bottom of the container.

7. Make a graph of your data. Choose your axes so that a linear relationship will verify Archimedes' principle. From your graph calculate the slope of the 'best fit' line.

8. Based on the physical analysis presented above, the slope of your graph should correspond to what quantity? Calculate the percentage error compared to the accepted value of this quantity.

Problem:

Note that this is not a very direct way to obtain the density of a fluid. On the other hand, a hydrometer uses a glass tube of fixed mass, which will sink to different depths in fluids of different densities. The scale on a hydrometer is marked directly in densities, instead of depth.

Make a hydrometer out of your test tube and use it to determine the density of the salt water solution at the front of the room.

Suggestion: Put enough weights in your test tube so that it floats vertically, is nearly sunk and has the water line at a convenient point on the y-scale. This line corresponds to a fluid density of 1.00 g/cm³.

Determine the correspondence between changes in y and the changes in fluid density. Imagine that the tube while maintaining the same mass were now floating 1 cm higher in the water. State your result: Each change in y by ______ corresponds to a change in density of _____ g/cm³.

Place the test tube in the salt water and read the water level on the y-scale. Determine the density of the salt water solution.

Question: The relationship verified in this experiment is rV = M. This looks like a relationship that would derive from the definition of mass density: r = M/V. Why is the relationship verified in this experiment not equivalent to the definition of mass density?