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A Rope Around the Earth (illustration)
FILE No.196Unsolved

A Rope Around the Earth

Difficulty ★★★about 5 minLogic PuzzleCircumferenceIntuition
A rope fitted tightly around the equator is lengthened by 1 m and lifted. How big is the gap?

Case Record

Assume the Earth is a perfect sphere and the length of the equator is about 40,000 km. A rope was wrapped tightly along this equator. The rope is lengthened by just 1 m and made to float at the same height all the way around above the ground.

Your question

About how large is the gap between the ground and the rope?

Case Files

Diagram

A rope fitted tightly round the equatorEarthEquator about 40,000 kmLengthen the rope by 1 m,and hold it at equal height all roundHow big is the gap?
Diagram

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

What is the relationship between circumference and radius?

Open hint 2

Circumference = 2π × radius.

Open hint 3

The radius increases by the increase in circumference divided by 2π.

Solve It

Your question

About how large is the gap between the ground and the rope?

Choose the answer closest to the truth.

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The Truth

About 16 cm

The answer is "About 16 cm."

When the circumference increases by 1 m, the radius increases by 1 ÷ (2π) ≈ 0.16 m. This is the same regardless of the size of the Earth.

1 mm or 1 m come from confusing the increase in circumference with the radius.

If the circumference grows 1 m, the radius grows…ItemDetailCircumference2πr (r = radius)Increase2π × (radius increase) = 1 mRadius increase1 ÷ 2π ≈ 0.16 m ≈ 16 cmNoteSame whatever the size of the Earth
Diagram of the solution

Key Points

  • Convert the added length into radius
  • The size of the Earth doesn't matter
  • Check with the formula

Why people get fooled

Because the Earth is so big you think 1 m changes nothing.

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