Puck's Girdle

Problem

Puck is a character in Shakespeare’s play Mid-Summer Night’s Dream, where, among other things, a queen gets to sleep with a country bumpkin with donkey’s ears. Puck’s a mischievous soul. But that’s irrelevant for the moment. At one stage he flies off and puts a girdle round the Earth in 40 minutes (Act II, Scene I).

How big is his girdle?

Puck magically sewed an extra 6m to his girdle and then made it into a circle around the earth. How far above the Earth would it be?

(Assume that the Earth is a sphere with no mountains etc.)

Teaching sequence

1. Discuss what a girdle is. How would you find the length of Puck’s original girdle?
2. Ask students to vote as to the size of the solution.
   How far out from the earth would the girdle be after 6 metres has been added?
3. Allow groups time to invent and carry out a method for solving the problem.
4. Summarise the various methods.

Extension

If Puck had repeated his two-girdle experiment on the Moon, how high above the Moon would the second girdle be?

Solution

The circumference of a circle is 2πr. We have used π as 3.14 in this problem. If we are talking about the Earth, then the radius is 6378km. So the circumference is 2π(6378). This is approximately 40053.84km. So this is the length of Puck’s girdle.

Now look at this the other way round. Start with a girdle of 40053.84+ 0.006 = 40053.846km. What is the radius of a circle with this as its circumference? Surely it will hardly lift off the Earth’s surface?

Let’s see. Now 2πr = 40053.846, so r = 40053.846/2π = 6378.00096. So the radius of the new girdle is 0.00096km more than the first one. That is 0.96m. Hang on! That’s almost a metre. Do you mean to say that adding 6m to the Earth’s circumference gives a circle with radius nearly a metre more than the Earth? That the girdle would stick up a metre all the way round the Earth?

Extension:

The fact is that it doesn’t matter how big the first circle is, the second girdle will always stick 0.96m above the surface.

Suppose that the original sphere had radius r. Then it’s first girdle would be of length 2πr. Now add 6m to that to give 2πr + 6, where we are now working in metres. What is the radius of a circle with circumference 2πr + 6? Well it’s simply (2πr + 6)/2π = r + 6/2π. So the increase is just 6/2π which is 0.96m - just under a metre. And that calculation is good for any value of r, whether it’s large like the radius of the Earth or small like the radius of a tennis ball. How about that?