Awesome Archimedes

Whilst working on a new project (about which more in future), I’ve had cause to look into the life of Archimedes, the ancient Greek mathematician, engineer, physicist, inventor and probable all-round good egg.

Archimedes is famous for shouting the phrase “Eureka!” – “I have found it!” – as he leaped out of his bathtub and ran through the streets naked. The word sums up that light-bulb moment when you figure something out.

Archimedes shouting "Eureka!" in the bath tub

The king at the time, King Hiero II, suspected that his crown was not pure gold, as intended, but had some silver substituted in its place. He wanted to know if this was the case.

Gold and silver have different densities – how much stuff is packed in how much space, or more formally, mass divided by volume. If you know the mass and volume of the crown, you can work out its density. You can then compare this with the density of gold (which was known). If they match, you know the crown in pure; if not, it has been doctored.

It was known at the time how to work out something’s mass – a pair of scales will suffice – but volumes of oddly-shaped objects were another matter. Archimedes figured out when taking his bath, that as you get in the water, you displace your own volume of water. That is, the water will rise up by the same amount of space that your body takes up.

All the King had to do was put the crown in a tub of water and figure out how much extra space the water took up to find the crown’s volume. Then he could calculate the density and find out if he had been deceived.

Archimedes also explained how levers work. The moment, or torque, provided by a lever is just the force applied times the perpendicular distance that force is from the pivot:

Moment = Force × Distance
Moment = Force × Distance

So if you apply 5N of force to a 20cm spanner (a fifth, 1/5, of a metre) to turn a screw, the torque you apply is:

Torque = Force × Distance

              = 5N × 1/5m

              = 1 Nm

With a lever, the moments either side of the pivot are equal. In the diagram below, a man applies a force of 10N to a lever, at a point that is 2m along the lever arm from the pivot. The lever is used to lift a load at the other end which is 1m from the pivot. I want to work out what force the lever applies to the load.

A lever is used to lift a weight

 The moment the man applies is:

Moment = Force × Distance

        = 10N × 2m

                 = 20 Nm

The same moment applies to the load:

Moment = Force × Distance

   20 Nm = Force × 1m

So the force applied on the load is:

     Force = 20 Nm / 1m

                         = 20N

By standing double the distance from the pivot than the load, the man has doubled his 10N force to 20N. If he had stood three times further, the lever would apply three times the force. Four times further, four times the force. Et cetera. Et cetera. Et cetera. With the force applied a large enough distance from the pivot, any weight could be lifted:

Archimedes levering the Earth
“Give me but a firm spot on which to stand, and I shall move the earth.” – Archimedes

Archimedes also developed a screw pump and reportedly focused the Sun’s rays to create a heat ray which set enemy ships on fire. You have to admit, he was one awesome dude.

 

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