Light weights

See how the weight of weights varies across the globe.

Age

16 to 18

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

The weight $W$ of an object on earth depends on the mass $m$ of the object and the force of gravity. The weight is usually given by the expression

$$

W = 9.8 m.

$$

However, the actual weight decreases the further you get from the centre of the earth. Newton worked out that weight can be measured more accurately as

$$ W =\frac{6.67428 \times 10^{-11}Mm}{R^2} N, \quad M = 5.972 \times 10^{24} \mbox{kg}. $$

Here $M$ is the mass of the earth, $m$ is the mass of the small object you are trying to weigh in $\mbox{kg}$ and $R$ is the distance from the centre of the earth in metres; $W$ is the weight in Newtons, which have units of metres kilograms per second per second.

In Olympic weightlifting the biggest competitors can sometime lift $200\mbox{kg}$ masses overhead. Sometimes weight lifting events take place in high altitude cities and sometimes at sea-level. The question that you are asked is this:

Does the variation in gravity provide a significant effect for weightlifters?

Something else to think about: How high in an airplane or rocket would you have to go before you could lift a $200\mbox{kg}$ mass overhead?

NOTES AND BACKGROUND

$G$ is called Newton's gravitational constant, which you can read about on Wikipedia.The universal law of gravitation expressed here gives extremely accurate predictions for the orbits of suns and planets. It is eventually superseded by the difficult theory of general relativity.

Teachers' Resources

Why do this problem?

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.

Possible approach

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.

Key questions

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]

Possible extension

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.

Possible support

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.

Altitude (km)	Weight (%, relative to 200kg)	Weight (kg)
0	100	200
10	99.69	199.37
50	98.45	196.90
100	96.93	193.87
500	85.98	171.95
1000	74.71	149.41
5000	31.39	62.78
10000	15.14	30.29
50000	5.84	11.67

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