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Shrinking world

Question: I once heard that if the Earth were shrunk to the size of a squash
ball or racketball, the planet would be smoother than a real squash ball.

Is this true? And if the converse happened and the ball were expanded to the
size of the Earth, how high would the mountains be?

Answer: To answer this intriguing question, we first need to establish the
scale factor we would have to shrink the Earth by, in order to reduce it to the
size of a squash ball. The Earth is 12,756 kilometres in diameter at the
equator, and a regulation squash ball has a diameter of 4.4 centimetres. This
means that to shrink the Earth down to the size of a squash ball, its size would
have to be multiplied by a scale factor of 3.45 × 10^-9.

To compare the smoothness of the two surfaces, we need to know the variation
of the surface—that is, the difference between the highest and lowest
points.

For a squash ball, this is a simple process, because there are very few areas
where the surface is higher than the average, but there are many small
indentations or depressions. Since these depressions are roughly 0.1 millimetres
in depth, the variation in surface height can be taken to be roughly 0.1
millimetres, or 10^-4 metres.

For the Earth, the lowest point below the surface is in the Mariana Trench,
which is 11,034 metres below sea level at its deepest point, known as the
Challenger Deep. The highest point is, of course, the summit of Mount Everest,
which is estimated at 8848 metres above sea level.

Therefore, the variation in the height of the Earth’s surface is 19,882
metres. If we scaled the Earth down to the size of a squash ball, using the
scale factor calculated above, the variation of its surface would be 6.86 ×
10^-5 metres, or 0.0686 millimetres. This figure is in fact about
two-thirds of the figure for the squash ball, so what your correspondent heard
is actually true—if Earth were scaled down to this size it would indeed be
smoother than the average regulation squash ball.

Now for the second part of the question. The lack of any raised areas on the
squash ball’s surface means that there are in fact mostly depressions or
indentations in the surface. So if a squash ball were scaled up to the size of
the Earth, there would be no mountains as such.

There would, however, be a lot of large craters. In fact, if we scaled up the
indentations in the ball’s surface we would end up with some immense depressions
that were almost 29 kilometres deep.

If these depressions were ocean trenches like those that are found on the
Earth’s surface, they would penetrate the 6-kilometre-thick oceanic crust, and
extend right through the Mohorovicic discontinuity— where the crust meets
the mantle—and well into the mantle beneath. These craters would not only
be deep but could be anything up to about 60 kilometres wide.

Tim Kelby

Irthlingborough, Northamptonshire

Answer: If the mass of the Earth were crushed down to the size of a squash or
racketball, then it would be dense enough to be either neutronium or a black
hole.

In the case of neutronium, the gravity at the surface would be in the order
of at least a million times the gravity you are feeling now, more than enough to
smooth out any irregularities in the surface. In the case of a black hole there
wouldn’t be a surface, just an event horizon—which would be smooth.

Expanding a ball to the size of the Earth is somewhat different. If we assume
that the ball is made mainly of carbon atoms and weighs 1 kilogram, then there
would only be around 22 atoms in each cubic centimetre. I believe that this is
actually less dense than the Earth’s upper atmosphere on the edge of outer
space.

Stephen Forbes

Leeds, West Yorkshire

The hole answer

Question: How is the hole in fine surgical needles produced?

Answer: The method that is used is generally similar to that used to make any
kind of tubing, including common copper domestic tubes, except that tubing in a
hypodermic needle is made from steel. This presents certain manufacturing
problems, in addition to the quality control needed to create such a
miniaturised product.

The first stage in tube manufacture is making a cylindrical billet, which for
hypodermic tubing would be a steel forging about 40 centimetres long and about 6
centimetres in diameter. A billet for copper tubing might be 1 metre long and 25
centimetres in diameter. The metal has to be forged to produce the necessary
directional grain. To make it workable, it is then softened by heating. The
billet is turned on a lathe and bored lengthwise to the required diameter.

Then the diameter of one end is reduced slightly for a length of a few
centimetres so that it can be entered into a die and gripped by a pulling
mechanism. The die is a thick disc of hard-surfaced, polished steel with a
radiused hole in the centre. Powerful machinery is used to draw the billet
through the die and reduce its outer and inner diameters by a very small amount.
The drawing is repeated many times with progressively smaller dies and regular
annealing.

The end result is a coil of fine tubing that is 50 metres long or more, with
the original hole proportionally reduced.

John Blundell

Bolton, Lancashire

This week’s questions

Strange tail: Why have rats got long tails?

Lucy Finlay (aged 9)

Ballynure, County Antrim

Steaming in: I have noticed that the amount of steam arising from frying pans
and saucepans increases rapidly in the seconds just after I have turned off the
flame under the pan.

Why is this?

Alex Aitkin

Canberra, ACT