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Physical Science: Session 2

A Closer Look: Scaling Analogies With Powers of Ten

apple

It’s not easy to imagine dividing something into ten pieces nine different times.

In order to make the difference in size scales more comprehensible, we can expand the size of the original object to something with which we are familiar. Let’s look at physicist Richard Feynman’s example of expanding a hydrogen atom to the size of an apple.

To make this calculation easier, we round off to the nearest power of ten. Feynman knew that an atom is a few angstroms (10-10 meters) wide. Let’s round off to 10 angstroms = 10-9 meters. An apple is about 10 centimeters (or 10-1 meters) wide. In order to make the hydrogen atom as large as the apple, we have to make it ten times bigger a total of eight times. In other words, we must expand it by a factor of 108:

The size of hydrogen atom (10-9 m) multiplied by the expansion factor (108) equals the size of an apple (10-1 m)

or:
10-9 m x 108 = 10-1 m

An easy trick when multiplying powers of ten is to simply add their exponents (the power to which ten is raised). In the above example, this shortcut gives us

10-9 m x 108 = 10(-9 + 8) m = 10-1 m

To rescale the size of the apple, so that we may compare it to our expanded hydrogen atom, we multiply the size of the apple by the same expansion factor:

The size of an apple (10-1) multiplied by the expansion factor (108) equals the size of earth (107 m or 10,000 km)
or:
10-1 m x 108 = 10(-1 + 8) = 107 m (or 10,000 km)
The diameter of the earth is approximately 12,000 km, so this is a good estimate.

Earth

Thus, if you expand a hydrogen atom to the size of an apple, the apple would expand to the size of the Earth. Having a student visualize the difference in scale between an apple and the Earth is more expressive than simply stating that an apple is 108 times bigger than an atom. This is the power of these kinds of analogies.

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