Powers of 10 and the size of things


$10^0$ m=$1$ m = 100 cm
$10^{-1}$ m=$0.1$ m = 10 cm
$10^{-2}$ m=$0.01$ m = 1 cm
$10^{-3}$ m=$0.001$ m = 0.1 cm

"The Scale of the Universe"

When comparing things of *vastly* different sizes, it's useful to use powers of 10.

The Scale of the Universe is an interactive viewer that compares things that have a typical length of $10^{x}$ meters.

When you click on an object, the size (longest length) of the object in meters, in exponential notation shows up at the lower right corner of the screen. For example, the Rafflesia flower shown is about 1 meter across. 1 m =$10^0$ m.

Open "The Scale of the Universe" site.

play around for a bit!, then answer these questions:

  1. What's a typical object at Goshen College (not something shown in the "The Scale...") which has a length of about:
    1. $10^2$ m = 100 m? 100 m: soccer field
    2. $10^1$ m? $10^1=10$ m: half a tennis court, or in the 'student power" assignment those of you who went up *all* the stairs figured a height of about 11 m, so one science building's height!
    3. $10^0$ m? $10^0=1$ m: mini-fridge!
    4. $10^{-1}$ m? $=\frac1{10} $m =10 cm: a grapefruit, a black squirrel
    5. $10^{-2}$ m = 0.01 m= 1 cm? a pea, a cheerio
  2. What's the size of a water molecule?


    $2.8\times 10^{-10} $ m.

  3. What's the size of one of the atomic nuclei shown?


    $6\times 10^{-15}$ m for the Chlorine nucleus: Larger than Helium, smaller than Uranium.

  4. How many times bigger is a water molecule than a nucleus? Take the water molecule size and divide by the atomic nuclei size...

    The ratio of their diameters (larger divided by smaller) is approximately (make sure both numbers are in the same units): $$ \text{ratio} = \frac{3\times 10^{-10}}{6\times 10^{-15}} $$ If you punch this into a calculator, it will tell you the answer is "46,666.6666666666666.....". But instead, let's

    Calculate like a physicist

    You can get an approximate number without a calculator, if you know some things about how exponents and fractions work: $$ \begineq \frac{3\times 10^{-10}}{6\times 10^{-15}} &=\frac 36\times\frac{10^{-10}}{10^{-15}}=\frac {3/3}{6/3}\times 10^{-10 - (-15)}\\ &=\frac 12\times 10^{-10+15}=\frac 12\times 10^5\\ &=0.5\times 100,000\\ &\approx 100,000 \endeq $$ Whoa! A water molecule is around 100,000 times the size of an atomic nucleus!

    I've used...

    • Separating or multiplying fractions
    • If you do the same thing to the top and bottom of a fraction, it doesn't change the answer!
    • I didn't use algebra, but remember that something similar happens with equations: Doing the same thing to both sides of an equation means the two sides will still be equal to each other. Solve the equation $1=\frac13 x$ for $x$: $$\begineq 1 &= \frac 13 x\\ 3*1&= 3*\frac 13 x = \frac{3}{3}x\\ 3 &= x \endeq $$
    • Exponents, division: $10^b / 10^a = 10^{(b-a)}$.
    • Exponents, multiplication:$10^b \times 10^a = 10^{(b+a)}$.
    • Exponents of 10 and moving the decimal point of 1.0
    In a chemistry class the answer you give (paying attention to significant figures) might be 47,000.
    In a physics class an answer to the nearest power of ten, in this case 100,000, is often good enough!

    3 rounds down, 4 rounds up!!: The power of ten is the logarithm (base 10) of a number. The logarithm of 4 is closer to the logarithm of 10 and the logarithm of 3 is closer to 1.

  5. What's one thing that surprised you?
    • marathon $\approx$ width of Rhode Island
    • That 15-25% of all land animals are ants.
    • That the smallest thing visible to a microscope is 50 picometers. [...still much larger than large molecules, like our soap molecules, let alone atoms?!]
    How to measure really small particles? Shoot thin "stuff" with the smallest particles possible! See Geiger, Marsden, Rutherford's "Gold Foil" experiment.