7356
/www/nrich/html/content/id/7356/
1
2011-05-09T11:36:05
<mdoxml version="1.0"><br></br>
<p>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 <br></br>
<br></br>
$$<br></br>
W = 9.8 m.<br></br>
$$<br></br>
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<br></br>
</p>
$$ W =\frac{6.67428 \times 10^{-11}Mm}{R^2} N, \quad M = 5.972
\times 10^{24} \mbox{kg}. $$ <br></br>
Here $M$ is the mass of the earth, $m$ is the mass of the small
object you are trying to weigh in $kg$ 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.<br></br>
<br></br>
In Olympic weightlifting the biggest competitors can sometime
lift 200kg 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:<br></br>
<br></br>
<span style="font-weight: bold;">Does the variation in gravity
provide a significant effect for weightlifters?</span><br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">Something else to think
about</span>: How high in an airplane or rocket would you have to
go before you could lift a 200kg mass overhead?<br></br>
<br></br>
<br></br>
<div class="framework">NOTES AND BACKGROUND<br></br>
<br></br>
$G$ is called Newton's gravitational constant, which you can read
about on <a href="http://en.wikipedia.org/wiki/Gravitational_constant">Wikipedia</a>.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eventually superseded by the difficult theory of general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. <br></br>
</div>
<br></br></mdoxml>
<mdoxml version="1.0"><p><span class="editorial">This problem is best tackled by using the formula given to you carefully. </span></p>
<p> </p>
<p><span class="editorial">Christopher from Sale Grammar sent us his working <a href="/content/id/7356/Christophersanswer.jpg">here</a>. He comments:</span></p>
<p> </p>
<p>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.</p>
<p> </p>
<p><span class="editorial">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?</span></p>
<p> </p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>Altitude (km)</td>
<td>Weight (%, relative to 200kg)</td>
<td>Weight (kg)</td>
</tr>
<tr>
<td>0</td>
<td>100</td>
<td>200</td>
</tr>
<tr>
<td>10</td>
<td>99.69</td>
<td>199.37</td>
</tr>
<tr>
<td>50</td>
<td>98.45</td>
<td>196.90</td>
</tr>
<tr>
<td>100</td>
<td>96.93</td>
<td>193.87</td>
</tr>
<tr>
<td>500</td>
<td>85.98</td>
<td>171.95</td>
</tr>
<tr>
<td>1000</td>
<td>74.71</td>
<td>149.41</td>
</tr>
<tr>
<td>5000</td>
<td>31.39</td>
<td>62.78</td>
</tr>
<tr>
<td>10000</td>
<td>15.14</td>
<td>30.29</td>
</tr>
<tr>
<td>50000</td>
<td>5.84</td>
<td>11.67</td>
</tr>
</tbody>
</table>
<p> </p></mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
<div>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.</div>
<div> </div>
<div><span style="font-style: italic;">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</span>.</div>
<h3>Possible approach</h3>
<div>This problem is ideally suited to students who are
familiar with the equation $W=mg$ but not familiar with Newton's
law of gravitation. </div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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? [<span style="font-style: italic;">There are no obvious
answers to these questions, but they are very motivating to
ask!</span>]</div>
<div> </div>
<div>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.</div>
<div> </div>
<h3>Key questions</h3>
<div>What is the difference between weight and mass? How might you
explain to someone that they are different [<span style="font-style: italic;">e.g. In space people are weightless;
they are dimensionally different</span>]?</div>
<div> </div>
<div>What units is mass measured in? What units is weight measured
in?</div>
<div> </div>
<div>How might we know the $W=mg$ cannot be always
correct? [<span style="font-style: italic;">e.g. In space
people are weightless</span>]</div>
<div> </div>
<div>How might we define the term 'significant' in the main part of
the question? [<span style="font-style: italic;">e.g. the weight
varies by less than 0.1N</span>]</div>
<h3>Possible extension</h3>
<div>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, <a href="http://nrich.maths.org/5941&part=">Earth Orbit</a> (very
difficult) or <a href="http://nrich.maths.org/5952&part=">Escape from Planet
Earth</a>.</div>
<h3>Possible support</h3>
<div>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. </div>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
Radius of Earth = 6378.1km. Assume R is radius of Earth + elevation
of city. If R is large then W is small. If R is small then W is
large. Here are the elevations of some Olympic Games host
cities:<br></br>
<br></br>
Athens - 338m<br></br>
St Louis - 142m<br></br>
London - 24m<br></br>
Mexico City - 3930m<br></br>
<br></br>
The highest cities in the world are La Rinconada, Peru (5100m) and
Wenzhuan, China (5019m).<br></br>
<br></br>
The difference in heights does not radically affect R.<br></br>
<br></br>
Let m = 200kg and define significant as weight varies by less than
0.01N.<br></br></mdoxml>
2
3
0
0
0
0
1
Light weights
See how the weight of weights varies across the globe.
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