What is nonlinearity in physics?

Answer 1

This question is best answered by example.

Suppose we have a square with a side length of two metres. Its perimeter is the length of four two-metre sides and that comes to eight metres.

Now suppose we triple the size - that is, make the sides three times as long. We now have a square with six-metre sides and its perimeter will be 24 metres - three times the smaller case.

The side-length and the perimeter of a square are LINEARLY related. If you change the side-length, the perimeter changes IN THE SAME PROPORTION. Double one, you double the other. Halve one, you halve the other.

Now consider the area of the smaller square. It is two times two - four square metres. How about the area of the larger square? Six by six comes to thirty-six square meters. But, if we trebled the side length - did we treble the area? No. Treble four is only twelve, not thirty-six. The area was double-trebled - three by three by four. That is thirty-six.

So, the side-length and area of a square are NOT LINEARLY RELATED. Changing the side-length of a square does not cause a proportional change to the area. The relationship is NON-LINEAR.

That example is in GEOMETRY, not PHYSICS. Here's examples in PHYSICS.

Consider a spring balance which is a pan suspended by a long weak spring. Putting weights in the pan causes the spring to stretch, and a pointer against a marked strip shows the weight in the pan. Noice that the marks are evenly spaced for 1KG, 2KG, 3KG, ... Six Kilorams drops the pan three times as far as two Kilograms. Springs, unless they are overloaded, show a linear relationship between load and stretch.

Now consider a tall thin can with a hole near the bottom. Fill the can with water and watch the water squirting out the hole. At first, the pressure at the bottom is considerable because there is a lot of water above it and so the water is forced through the hole at a good speed. However, as the level of water in the can drops, so does the pressure and the speed of the water-jet dies away too. Now we could ask ourselves, "When the water is half-way down, is the water spurting out at half the speed it started at? or "If we had a can ten times as high, would water squirt out the bottom ten times as fast?"

If the answer to these questions is "Yes", then the height of water and the speed it squirts out of a hole have a linear relationship. We won't go into the mechanics of this situation. We'll just have to be happy with my claim that this is an example of NONLINEARITY. Would you believe that the rate of flow of water in the hole depends on the square-root of the height in the can?