This problem provides an interesting context in which to
engage with mass, weight and gravitation. It is a good
mathematically straightforward question around which to base more
general discussions concerning mechanics and physics. It gives a
simple introduction to the use of Newton's law of gravitation and
students are likely to want to know the answer once the question is
posed. It tackles the fundamental physical observation that weight
and mass are very different sorts of attribute: mass is intrinsic
to an object, whereas weight is determined relative to another
object.

Note that this problem is
likely to raise several issues concerning familiar concepts. For
example, does the height of a city added to the radius of the earth
actually give the distance from the centre of the earth? As a
teacher, please note that you are not expected to be able to give
definitive answers to such questions! Such questions can be left
open or solved collectively to the best of the ability of the
class.

This problem is ideally suited to students who are
familiar with the equation $W=mg$ but not familiar with Newton's
law of gravitation.

As with many ideas in mechanics it is worth having some
discussion on the physical ideas before launching into the
mathematics; the same is true here. Discuss weight and mass as a
group to be sure that everyone understands the difference between
the two.

It is worth making the observation that physical laws such as
Newton's law of gravitation are discovered in part by observation
and in part by mathematical analysis. This might be a good moment
for students to note the beauty of the mathematical equations
underlying physics: an inverse square law is rather beautiful; why
is the power exactly 2; why does it seem to work everywhere in
space? [There are no obvious
answers to these questions, but they are very motivating to
ask!]

Once students are ready to begin to answer the main question
they will realise that they will need to make approximation,
estimations and to use data, which will be readily available
online. Allow students to make this realisation for themselves,
rather than to provide them with numbers in advance. Some students
might struggle with the lack of 'precision' with which the question
is posed. Encourage them to define clearly a meaning of the word
'significant' so that a meaningful analysis can take place. Some
students will wish to think about this themselves; others in small
groups - either is fine, but a good answer will involve a brief
explanation of the modelling steps and assumptions made.

What is the difference between weight and mass? How might you
explain to someone that they are different [e.g. In space people are weightless;
they are dimensionally different]?

What units is mass measured in? What units is weight measured
in?

How might we know the $W=mg$ cannot be always
correct? [e.g. In space
people are weightless]

How might we define the term 'significant' in the main part of
the question? [e.g. the weight
varies by less than 0.1N]

Hopefully interested students will wish to make other
calculations to answer questions which come to mind. If they wish
to solve another 'problem' then please see, for example, Earth Orbit (very
difficult) or Escape from Planet
Earth.

Team work should be sufficient to find an answer to this
problem; suggest that those struggling work together and discuss
their difficulties with others. You could give the hint that height
above sea level combined with the radius of the earth can give the
distance from the centre of the earth.