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# Light Weights

##### Age 16 to 18Challenge Level

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 It is eventually superseded by the difficult theory of general relativity.