Problem 1 (The Rope-Around-the-World Problem)

A rope is wrapped around the earth at the equator three feet in the air. If we assume the earth is a perfect sphere with radius 8,000 miles and if we drop the rope to the ground, how much slack will there be in the rope ? You might make a guess before you solve this problem.

Solution: If we call R the radius of the earth (in feet) then the circumference of the rope around the earth before it is dropped to the ground is

C_rope = 2 (pi) (R + 3 )

and the circumference after it is dropped to the ground is the radius of the earth, or

C_earth = 2 (pi) R

Hence, the slack is the difference between the two circumferences, or

C_rope = C_earth = 2 (pi) (R + 3) - 2 (pi) R = 2 (pi)(3) =6 (pi) = 18.18 feet

The interesting thing about this problem is that the amount of slack does not depend on the radius of the earth. In fact, if a rope 3 feet above a basketball is dropped to the basketball, you would have the same slack of 6 (pi) = 18.85 feet. Also, if you are jogging around a 400 meter track with a friend and if you are running on the outside 3 outside your friend, then on every lap you will run an additional 2 (pi) (3) = 6 18.85 than your friend (since the only time you run further is on the two ends of the track which can be collapsed into a circle).

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Last modified on Wednesday, March 17, 1999