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FATAL ATTRACTION

How much are you physically attracted to your partner?

Have you ever woken up in bed with a numb arm or leg? The blood flow to the affected limb is constricted and leaves it feeling deadened. So you change position, the blood circulates again, and you go back to sleep.

The weight of a dead leg or arm seems enormous as you drag it into a more comfortable position. But how much do your individual body parts actually weigh? Your whole body weight is easy to measure: you just step onto some scales. Weighing a leg, an arm or even your head is more challenging, assuming that your body is to be left intact in the process.

Here is a practical method of working out an approximate value for the weight of your head while it is still connected to your body. You’ll need a bucket large enough to immerse your whole head in, and a kitchen measuring jug. A bathroom towel might make the experiment more pleasant. It is best to do it in the garden, though be careful not to alarm the neighbours.

Fill the bucket to the brim with water and place it on the ground. Kneel next to the bucket, take a deep breath and gently immerse your head in the water, up to your neck. Water will spill out of the bucket, but this is quite normal.

When you’ve surfaced, the bucket will no longer be full. In fact, the amount of water that has overflowed will be exactly equal to the volume of your head. You can get a good enough measure of that volume by topping up the bucket with water using the measuring jug, until it is again full to the brim.

I just did this; you might notice the wet pages. The volume of my head I worked out to be precisely 5 litres. Now, I wouldn’t like you to think that I’m being big-headed about this, but that makes my head the same size as one of those plastic petrol containers you keep in your car boot in case of emergency.

Hopefully you’ve also done the experiment and will now know the volume of your own head in the units of your measuring jug – litres, pints or fluid ounces. If the stuff in your head had the same density as water, the weight of your head would be equal to the weight of the same volume of water. A litre of water weighs 1 kg, so the weight of your head in kilograms would be the same as the number of litres of water. In my case (and for most adults) that’dbe5 kg.

Despite appearances, your body is mostly water. But compared with other parts of the body, a head contains quite a lot of bone relative to soft tissue, and so is going to weigh more than water, maybe 10–20% more. So just increase your initial head weight estimate by this amount. That brings it up to 6 kg.

In scientific terms, weight – whether of your head, your whole body or anything else – is a force that pulls between the mass of an object and the mass of the Earth. If you step onto old-fashioned bathroom scales that work by compressing a spring, the force that is your weight compresses the spring as surely as if you had squeezed it with your hands. What gives rise to that force is, of course, gravity.

Sir Isaac Newton worked out that the strength of the gravitational force pulling two objects together increases in proportion to both masses, but decreases rapidly as the distance between the masses increases. He showed that the force was quartered for each doubling in distance.

We can write this as a formula:

Force=6.67e - 11×Mass1×mass2 ×Distance²

That number, 6.67e ? 11, is called the universal gravitational constant. (The number written out in full is 0.0000000000667, but it is very tedious to write down all those zeros, and it’s easier to use the shorthand built into calculators and used in computer programming languages. The –11 after the e indicates that the digits ‘667’ are preceded by a decimal point and 11 zeroes. We can use the same notation to save writing out very large numbers in full: 4.567e9, for example, is 4,567,000,000, the age of the Earth in years.)

The first person to devise a method that could be used to get a very accurate value for the universal gravitational constant was Henry Cavendish. In 1798 Cavendish, reputedly a very shy man who may have had Asperger’s syndrome, set up an experiment using an instrument called a torsion balance.

The torsion balance is a wooden rod suspended at its midpoint by a silk thread. Weights are hung from each end of the rod, a bit like the kind of mobile that people hang above a baby’s cot. The delicate mechanism was sealed in a room so that it would not be disturbed by draughts.

Cavendish moved large balls of lead near to the outside walls of the sealed room, and they caused the suspended rod to twist very slightly because of the force exerted by the lead balls on the weights hanging on the torsion balance. The amount of twist can be used to work out the force and hence the universal gravitational constant.

That same gravitational force pulls on your body whenever there is a nearby massive object, for example a ship, a mountain or another human being. Unlike the pull from the Earth, those forces act along a line between you and the other object.

For example, when you stand right next to your partner, how much are you pulled together by the gravitational force between your bodies? Let’s make it intimate – say you and your partner are standing face to face, just lightly touching.

It’s simple to work out the strength of your attraction to your partner. You’ll need some numbers to plug into the gravitational force formula.

Say that Object1 is you. What is your mass? In numbers, it’s the same as your weight in kilograms. This is just an estimate, so use a nice round number, say 100 kg. Next you need the mass of Object₂. That’s your partner, and it’s up to you to negotiate a value to insert here, but the calculation will be easier if you use 100 kg again. It may, of course, make other parts of your life more difficult.

Lastly, to use the formula you need a value for the distance between your bodies – and this is the interesting bit, because no single distance is correct. The distance at the points where your bodies touch, say, umm… chest to chest, is almost zero, but the distance between your buttocks and your partner’s buttocks might be anything up to 60 cm.

The proper method of dealing with a problem like this is to imagine both bodies cut into very thin vertical sections, as if you had been put through one of those little utensils for cutting a boiled egg into slices.

Each slice would look like a paper cut-out silhouette of your body. You would use the area and thickness of each slice to get its mass, and then work out the distances between every slice of your body and every slice of your partner’s body.

The gravitational forces between every pair of slices can then be worked out from the formula, and added up to give the total force of attraction. The mathematical process is called integration, and is easy to do on a computer, but too complex to do in your head.

What we need is a rough and ready method which will tell you quickly how much you are attracted to each other. He/ she may be waiting impatiently to know the answer.

A simple solution is to assume that for you and your partner, all of your mass is concentrated at a single point in the middle of your body, say halfway between your belly button and the small of your back. That might be around 15 cm in from your front, making the effective distance between your body masses 30 cm.

And now you can run the formula. These are the values we need to plug in: Mass₁ = 100 kg, Mass2 = 100 kg and Distance = 0.3 metres. So your mutual attraction will be:

That’s a force of about 0.0000074, and it is measured in units called newtons – after the great man himself.

Newton was involved in just about every field of mathematics and physics on which our modern lives are based. He formalised our understanding of gravity and force, coining the term ‘gravity’ from the Latin word gravitas, which means ‘weight’. He invented differential calculus, and designed a type of reflecting telescope that is still called a Newtonian telescope.

Strangely, he got his knighthood not for his scientific advances but for his work as Warden of the Royal Mint. He stabilised the currency by linking the value of silver coins to the value of gold – a kind of precursor to the Gold Standard, used off and on until the Second World War.

He probably wasn’t much interested in the kind of inter-body attraction we are calculating. He seems to have had only one girlfriend, whose affections dimmed as he immersed himself in science.

One newton is about the force needed to lift a 100-gram weight, so the force between you and your partner is only about the force needed to lift 0.000074 grams. Maybe Newton had already calculated that his force was too small to hold people together.

It is a measurable force, but you would be well advised to rely on the force of your personality, or other attractions, to keep your partner close.

SPEAK GEEK

A MAN WHO WEIGHS 100 KG AT THE NORTH POLE WOULD WEIGH ONLY 99.65 KG AT THE EQUATOR.

Your net weight is the result of two forces. The first force is gravitational: it pulls your body mass downwards, towards the centre of the Earth. The second force is upwards, and is caused by the rotation of the Earth constantly trying to fling your body into space like a stone on a string being swung round your head.

At the exact point of the North Pole, the Earth experiences no rotation, and so there isn’t any upward force. On the other hand, the Earth’s rotational speed at the equator is at its maximum – about 1,500 kph – and a small amount of the Earth’s gravitational pull is cancelled out by the upward ‘flinging’ force. Hence the weight loss.

Beat that for a diet, Professor Atkins.