Quadratics

1.5 Galileo’s Mistake

Lesson materials located below the video overview.

It turns out, as we shall see, that any curve given by a formula of the form \(at^{2} + bt + c\)  has a U-shape. And this U-shaped curve appears many places in nature.

 

Galileo noticed shape of a rope or a chain hanging between two poles is U-shaped and deduced that it too must be given by a quadratic formula. (We see this with the shape of power lines, the shape of ropes that surround sculptures in art museums, and so on.)

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We have the means to find formulas for quadratic data!

 

ACTIVITY:

1. Hang a piece light-weight chain on a white-board.

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2. With a ruler, draw a horizontal line and mark of evenly spaced positions. Measure heights from that horizontal line to the chain and collect a sequence of data values.

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3. Use your data to find a quadratic formula for the hanging chain. (Look at first differences and second differences.) Be honest about what you notice.

 

 

DO NOT READ THIS NEXT SECTION UNTIL YOU HAVE ACTUALLY TRIED THIS ACTIVITY FOR YOURSELF AND LOOKED AT YOUR OWN DATA …. THERE BE SPOILERS!

There will no doubt be inaccuracies in your data and you will find that the first second differences you calculate are not constant. Go back and try to do more refined and accurate measurements.

 

But you will find that the double differences are still not constant.

 

… And try as you might, with more and more accurate measurements you will fail to see constant double differences!

 

It turns out that Galileo was in error about the shapes of hanging chains: they do not follow a quadratic formula. Trust what your data is telling you!

 

The shape of the curve given by a hanging chain became known as a catenary curve (from the Latin catena for “chain). It wasn’t until 1691 that mathematicians found a precise formula for this curve. (It is very different from a quadratic formula!)

 

This activity provides a powerful tool for testing for quadratic curves.

EXTENDED ACTIVITY: Is the St. Louis Gateway Arch in the shape of a quadratic curve? Find out by making measurements on a photograph of the arch.

EXTENDED ACTIVITY: Is a semi-circle given by a quadratic formula? Find out by tracing a pot lid on a piece of paper and making measurements with a ruler.

EXTENDED ACTIVITY: Is the loop of a bouncing ball quadratic? Find a time-lapse series of photographs of a bouncing ball (on the internet or use a digital camera to take photographs of a ball every 0.1 seconds, say). Is the height of the ball following a quadratic formula?

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