Additional Math Pages & Resources

Wednesday, July 7, 2010

Relative Spherical Density

I noticed the blog yesterday primarily dealt with 3-dimensional objects known as spheres, rather than 2-dimensional shapes known as circles. Titling it "Going around in circles" was misleading. Sorry. But let's not give up on the subject yet. Did you notice the items weighed different amounts even if they were of similar sizes? That means their density varied. Today we calculate relative spherical density.

Here are a few spheres that are NOT sporting equipment. The one on the left is made of solid copper. The one on the right is foam rubber - I squeeze it to give my fingers some relief from too much typing.


Following yesterday's formula, here are the relevant dimensions:

Copper Sphere 5cm diameter  600g weight  &  Rubber Sphere 7cm diameter  30g weight.

We can compare these two objects by multiplying and dividing their dimensions.

In Excel Math we teach elementary school kids how to demonstrate comparisons - for example:

Q1. How much larger is the rubber sphere than the copper sphere?

A1. We divide 7 by 5 and learn that the rubber sphere is 1.4 times as large ( 7cm ÷ 5cm = 1.4 )

Q2. How much larger is the copper sphere than the rubber sphere?

A2. It's ( 5cm ÷ 7cm = .7 ) or seven-tenths as large.

Q3. How much heavier is the copper sphere than the rubber sphere?

A3. ( 600g ÷ 30g = 20 ) so the copper sphere is 20 times heavier than the rubber.

Notice that in all these answers the units disappear. A comparison does not have units!

Q4. For a given volume, how much heavier is copper than foam rubber?

A4. This is much more complex! The formula for volume of a sphere is 4/3 π r³. 
In words, it's four thirds times pi times the radius cubed (multiplied by itself 3 times)

The radius of a sphere is half the diameter so we could end up with something like this:

(4 ÷ 3) x 3.14 x ( d ÷ 2) x ( d ÷ 2) x ( d ÷ 2)

We have to calculate both copper and rubber, so to save time we can do this 4/3π stuff only once and get a "constant value" to use whenever we calculate spherical volume. (4 ÷ 3) x 3.14 = 4.19

Copper
4.19 x (2.5 x 2.5 x 2.5) = (4.19 x 15.625) = 65 cubic centimeters
Rubber
4.19 x (3.5 x 3.5 x 3.5) = (4.19 x 42.875) = 180 cubic centimeters

Excuse me, what was the question again? Oh yes, relative density (mass or weight per unit of volume)

Copper
600 ÷ 65 =  9.23 grams per cc
Rubber
  30 ÷ 180 = .1666 grams per cc

Now we can calculate the answer:

9.23 ÷ .1666 = 55      Copper is 55 times heavier than rubber for the same volume 
(any volume is 55 times heavier - the units disappear).