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Circumference and Radius
Name: Devon H.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
I have heard that if a string were tied tightly around
the earth's equator it would be 25,000 miles long (roughly). If the
string was then cut and six feet was added to the string, and then the
string was stretched an equidistant height off the ground, the string
would be one foot off the ground. I know this is hypothetical, and
involves many generalities (among them the circumference of the earth)
but the math seems to play out.
25,000 x 5280 feet / 2pi = radius of earth in feet
((25,000 x 5280 feet)) + 6 feet) / 2pi = new radius
If we subtract original radius from new radius we get 1 foot.
It does not seem logical. How can I explain this to my students?
Replies:
Devon,
You might try explaining it in general before mentioning the Earth.
Circumference equals 2pi times radius. If you add one foot to the radius,
circumference increases by 2pi feet. If you add one inch to the radius,
circumference increases by 2pi inches. It works for all circles.
Another way you may find useful is percentages. Calculate the percent
increase for radius. Calculate the percent increase for circumference. You
will find them to be equal. Compared to the radius of 21 million feet, one
foot is not a large amount.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
The math is all there in your question. The difficulty is that such a small
difference in circumference should not seem to make such a big difference in
radius.
However, the proportions are preserved. It is true, six additional feet is
not much of a difference when compared to the large circumference of the
earth. By the same token, though, one additional foot is not much of a
difference when compared to the large radius of the earth. Circumference
and radius are both linear dimensions, and are proportional to each other.
Interestingly, it does not matter what the initial circumference and radius
of a circle are, increasing the circumference by six feet will increase the
radius by about one foot. You can start from a point (radius =
circumference = zero), a circumference of two feet, or the orbit of the
earth around the sun. It does not matter. A change of X in the
circumference results in a change in the radius of X / (2pi).
Richard E. Barrans Jr., Ph.D.
PG Research Foundation, Darien, Illinois
If you make a relatively small change in the circumference, you would
expect to see a relatively small change in the radius, and this is in
fact what you see, because the radius changes from 21008452.49 to
21008453.44.
Tim Mooney
If the circumference of the earth is taken as 25,000x5280 feet the radius
is:
C/2*pi = 132000000.0 / 2*pi = 2.10084524881302 x 10^7 feet
If the circumference is increased to 132000006.0 / 2*pi = 2.10084534430598 x
10^7 feet
The increase in radius then is 2.10084534430598 x 10^7 - 2.10084524881302 x
10^7 =
9.54929597618559 x 10^-8 feet.
Vince Calder
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Update: June 2012
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