# Light weights

## 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

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?

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 It is eventually superseded by the difficult theory of general relativity.

## Student Solutions

This problem is best tackled by using the formula given to you carefully.

Christopher from Sale Grammar sent us his working here. He comments:

Using the values of 1956N and 1975N for the extremities of the earths surface we can see that the range of values would be 19N which equates to around 2kg. so, if a weightlifter was to do some weightlifting at the bottom of the Arctic, and then the top of Mount Chimborazo, they would feel a slight difference. However, when it comes to the main aim of the olympics, winning, it is clear that of course, everyone will be under the same stresses, so in terms of competition, everything will be fine. And there are other considerations at various altitudes too, such as air quality and density, and there are some serious considerations to be had at the bottom of the arctic. So in conclusion, yes, it would make a difference to the appeared strength of the weightlifter, but only by about 2kg.

Patrick from Otterborne included a table with his solution, showing the weight varying with altitude. How high would you have to go to lift 200kg?

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 |