Buoyancy of Floating Cylinders

Objective

The goal of this project is to measure how the tilt angle of cylinders floating in water depends on the aspect ratio (length/diameter) of the cylinder.

Introduction

If you place a wooden disk in water, it floats 'face up,' i.e., with the circular cross-section parallel to the surface of the water. However, if you place a long wooden cylinder in water, it floats with the circular cross-section perpendicular to the surface of the water (see Figure 1).

Figure 1. Illustration of a floating disk (A) and a floating cylinder (B).

If you think about it, a disk is a cylinder, too. A disk is just a very short cylinder, and 'disk' is just a special name for this type of cylinder. How short does a cylinder need to be before we call it a disk, or is there something more to it? A coaster for a hot cup of coffee certainly fits our concept of a disk. A ceramic coaster might be almost a centimeter tall and ten centimeters in diameter. However, we wouldn't call a one-centimeter length of pencil lead a disk, we'd call it a cylinder. That's because the diameter of the pencil lead is only 0.05 cm (0.5 mm). So apparently we consider both the length and the diameter of a cylinder when we're deciding whether or not it's a disk.

A handy way to consider both numbers at once is to use a ratio. For example, if we can use the ratio:

The coaster has an aspect ratio of 1/10, and the pencil lead has an aspect ratio of 1/0.05 or 20. So perhaps what we mean by a disk is a cylinder with an aspect ratio <>

Does the way a cylinder floats also depend on its aspect ratio? Since the disk floats face-up, but a longer cylinder floats with the circular faces perpendicular to the surface, does that mean that there are cylinders with intermediate aspect ratios that would float at intermediate angles? Do an experiment to find out!

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• aspect ratio.

More advanced students should also study:

• buoyancy,
• volume,
• mass,
• density,
• center of gravity,
• center of buoyancy,
• righting moment.

Questions

• What is the difference between the center of gravity of the cylinder and its center of buoyancy?

Bibliography

• Here are some good background resources on buoyancy:
  
  • Junior Engineering, 1997. "Buoyancy," Utah State University [accessed December 29, 2006] http://www.engineering.usu.edu/jrestate/workshops/buoyancy/buoyancy.php.
  • Wikipedia contributors, 2006. "Buoyancy," Wikipedia, The Free Encyclopedia [accessed December 29, 2006] http://en.wikipedia.org/w/index.php?title=Buoyancy&oldid=97455729.
• For more advanced students interested in the forces involved in determining the tilt angle, this article provides a good introduction to the concept of a righting moment:
  Shepard, J., 2003. "Stability: Capsizing Can Turn Your Whole Day Upside-Down," About: Powerboating [accessed January 19, 2007] http://powerboat.about.com/library/weekly/aa061903a.htm.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• wood dowels,
• small handsaw for cutting dowels to various lengths, notes:
  • You may even want to visit a hobby store to purchase a small aluminum miter box with a razor saw to go along with it. With a miter box to hold the dowel in place, it is easier to make perpendicular cuts.
  • X-acto and Zona are two quality brand names to look for. Both companies offer miter boxes and saws.
• dish pan or large plastic box,
• water,
• food coloring,
• metric ruler,
• protractor,
• pencil,
• blank paper,
• graph paper,
• calculator.

Experimental Procedure

1. Use a hand saw to cut cylinders of various lengths from a long piece of dowel. You'll need to experiment and figure out what range of lengths you need in order to see different tilt angles in water!
2. Measuring the aspect ratio of your cylinders is easy. Just measure the length (in cm) and the diameter (in cm), then divide the length by the diameter.
3. Measuring the tilt angle of the floating cylinders is a bit trickier. Here's how:
   • Carefully float the cylinders in water with food coloring added.
   • Allow the cylinders to float, undisturbed, for several hours.
   • The dye from the food coloring will stain the underwater portion of each cylinder. After a few hours, there will be a distinct line of dye marking the water line on each cylinder.
   • Remove the cylinders from the water and allow them to dry.
   • Note: if you like, you can also float the cylinders in colored liquid Jello, then allow it to set in the refrigerator. (You may need to occasionally nudge the cylinders away from the edge of the dish.) The food coloring in the Jello will stain the submerged portion of each cylinder.
4. Use the following steps to measure the tilt angle of each cylinder:
   1. Using a pencil and ruler, draw a straight line on a piece of paper.
   2. Place the dyed cylinder over the straight line, and tilt it until the dye line on the cylinder is parallel with the line on the paper (Figure 2A).
   3. Holding the cylinder in place, place a ruler against the cylinder at the same angle. (Figure 2A).
   4. Move the cylinder out of the way and use the ruler to draw a straight line that intersects with the original line on the paper.
   5. Use your protractor to measure the angle between the two lines (Figure 2B).
   6. To keep track of your measurements, we suggest that you use a separate sheet of paper for each cylinder. Label each angle drawing with the length, diameter, and aspect ratio of the cylinder.

      Figure 2. Measuring the tilt angle of the dyed dowel.

5. Make a table of your results like the one below:

   Length (cm)	Diameter (cm)	Aspect Ratio (length/diameter)	Tilt Angle (°)

6. Make a graph of your results by plotting tilt angle (y-axis) vs. aspect ratio. Over what range of aspect ratios does the tilt angle change?

Variations

• Try dowels of different diameters, but with the same density. For each diameter, cut dowels of various lengths and measure their flotation angles. Make a graph of flotation angle (y-axis) vs. length of the cylinder (x-axis). Use a distinct symbol for each diameter. How do the graphs compare for each cylinder diameter? Now make a graph of flotation angle (y-axis) vs. aspect ratio of the cylinder (length/diameter). Use the same symbols as before. How do these graphs compare?
• Try cylinders with different densities. Is the relationship between flotation angle and aspect ratio the same or different? Can you find cylinders made of different materials but with the same density? Do they have the same relationship between flotation angle and aspect ratio?
• For more advanced students: can you come up with an explanation of the physics behind the tilt angle vs. aspect ratio relationship? Can you figure out an equation that describes the relationship between tilt angle and aspect ratio? The following article will be a useful reference: Gilbert, E.N., 1991. "How Things Float," The American Mathematical Monthly, 98 (3, March): 201–216.

Sources

• "How Things Float," The American Mathematical Monthly, 98 (3, March): 201–216.

Related Posts by Categories