If our bird from last week were enclosed in a glass cage, with air entering and exiting only from above, the scale would still read 1.030 grams as it fluttered around, because all the reaction from the flapping action that keeps it aloft would affect the bottom of the cage. In a normal cage, open on all sides, part of that reaction would affect the table it was resting on or the floor of the room, so the scale would read a little less than 1.030 grams—just a little, in principle, because the reaction would still affect mostly the bottom of the cage (although the precise calculation would involve considerations of fluid dynamics, which is one of the most complex branches of mechanics).
The case of the fish in the fish tank is different: at the moment of the impulse that gives rise to the jump, the scale will read a little more than 1,030 grams due to the reaction in the water, which affects the base of the fish tank, since the jump is upwards; but while the fish is in the air, the scale will read 1,000 grams (it may even read a little less, due to the rebound effect), although only for a moment, because as soon as the fish falls back into the water it will read a little more than 1,030 grams for a fraction of a second, due to the impact, and then quickly stabilise again at 1,030 grams.
In the case of the iron ball, when it rests on the bottom of the fish tank, the scale reads 2,000 grams. When you put your hand in, Archimedes’ principle and Newton’s third law combine to make the scale indicate an increase in weight equal to the volume of water displaced; if your hand displaces half a litre of water, the scale will read 2,500 grams; but the moment you start to lift the iron ball, the scale will read considerably less. How much less and why?
The equivalence between the envelope to be drawn without lifting the pencil from the paper and the bridges of Kaliningrad lies in the fact that in both cases we have two nodes with an even number of concurrent paths and two others with an odd number. The upper vertex of the envelope does not count as a node of the graph, since we can replace it with a single curved line.
Tell your grandmother
If you find it abusive to equate a broken line – the flap of an envelope – with a curved line, it is probably because you are very clear about what a curve is. Or so you think. Einstein said that if he was not able to explain something to his grandmother, that meant that he did not fully understand it either (that is why he never accepted quantum mechanics: can you imagine Einstein’s grandmother saying: “Don’t talk nonsense, Albert, how can a cat be alive and dead at the same time?”). So try to explain to your imaginary grandmother – or to the real one, if you still enjoy her attention – what a curve is and you will see that it is not so simple. It is not about giving an approximate explanation, but a precise definition that is applicable to all curves.
The ancient Greeks gave several definitions of curves. The best known is the one that says that a curve is the intersection of two surfaces (which includes a straight line, considered a curve of zero curvature, as an intersection of two planes). That is why we call the circle, the ellipse, the parabola and the hyperbola “conics”, because they can be obtained as intersections of a cone with a plane at different angles.
Another classic definition is that of “geometrical locus”: the curve is the place occupied by the points that meet a certain condition; thus, a circle with radius R and center C is the geometrical locus of the points of the plane whose distance to point C is R.
The development of analytical geometry in the 17th century allowed the concept of geometric locus to be expanded: curves are the graphic representations of algebraic functions, or in other words, the geometric locus of points whose coordinates are solutions to an equation with two unknowns. But not all of them, since the graphs of certain functions are sets of disconnected points or lines that we would never call curves. And things became even more complicated when, at the end of the 19th century, the Italian mathematician and philosopher Giuseppe Peano (1858-1932) revealed a “monstrous” curve (others called it “pathological”) that, in the limit, becomes a compact square: a leap from the first to the second dimension more appropriate for a science fiction story than a geometry treatise. But that is another article. Or several.
Returning to the familiar, beautiful and not at all pathological cones, which of them—and how—could you generate with a ball and a flashlight?
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