Area Volume

Ratios Matter

An Exploratorium Teacher Institute Draft Activity

area volume cubes


Styrofoam cubes of different sizes can be charge on one surface and then stuck to a wall. Whether they stick or not is a function of the ratio of the area of one charged surface to the mass of the entire volume.


Styrofoam cubes:  For example sheets of 1" (2.5 cm) thick Foamular pink home insulation by Dow-Corning available at Home Depot.
Ruler pencil and saw (to mark and cut foam.
Hot melt glue gun (to assemble cubes.)
Wool cloth.
Wall or cardboard sheet


Mark a 1 inch, 2.5 cm, square on the surface of the 1 inch thick foamular and saw it out.
Mark two 2 inch, 5 cm, squares.
Mark three 3 inch, 7.5 cm, squares.

area mark
1inch, 2 inch and 3 inch squares marked and ready for cuts.

Mark four 4 inch,20 cm, squares.
Saw them all out. Clean the debris

cube parts
Use the hot melt glue gun to glue the matching squares together to make cubes.
make the cubes with holt melt glue

See the top image for the assembled cubes.

To Do and Notice

Lay out the wool cloth on a hard flat surface.

Put one smooth face of the single cube (1" or 2.5 cm cube) on Ute cloth and rub it back and forth 20 times about 10 cm per stroke.

Place that face against a wall and notice that it sticks to the wall.
Next rub the 2inch cube the same way and try to stick it to the wall.

2 inch cube stuck to wall
Then try the 3 inch and 4 inch cubes.
Notice that the 1 inch cube usually sticks to a wall, the 2 and 3 inch cubes sometimes stick.
And the 4 inch cube hardy ever sticks.


What's Going On?

When you rub the styrofoam on the wool the foam becomes negatively charged on the rubbed surface.

When the charged cube is brought near a wall it induces an opposite charge at the surface of the wall by induced polarization.

If the force of electrostatic attraction is large enough to create a friction force with the wall that is large enough the cube will stick to the wall even though it is being pulled down by gravity.

The electrostatic attractive force depends on the area of one face of the cube. The area is proportional to the square of the side length of the cube.

The force of gravity depends on the volume of the cube.The volume is proportional to the cube of the side length of each cube.

So the 4 inch cube has 4^2 = 16 times the area and 4^3 = 64 times the mass of a single cube.

The ratio of the area to the mas is 16/64 = 1/4.

It is much harder to stick a larger cube to the wall.

So What?

The behavior of the ratio of surface area to volume is extremely important at the nano scale where surface effects assume a much greater importance.


Whether the cubes stick to the wall depends on the roughness and composition of the wall.

Scientific Explorations with Paul Doherty

© 2010

25 October 2010