|Units calculation (units, names and values - small calculations with units - (measuresand appareils de mesure) - Disguised percentages - Slopes, degrees and trigonometry (clock and sinusoidal curbs, lunar tides and alternative current)|
Percentages and trigonometry.Percentages and slopes and degrees: their definitions can be misleading; one perfectly understands an increase of 10% : it is the tenth of the basic value. For 100 €, 10% make 10 € and one finally get 110 €. Beyond the 100%, the brain panics. Example, the cost has increased by three hundred percent : one retain the figure of three hundreds, which appears enormous, but one mentally does not make division per hundred. In fact, the cost is four times higher. 100% of increase double the value, 200 % triple it.
The slope of the funicular is 80%: we imagine immediately a quite vertiginous vertical, that the cameraman don't middto to amplifying while banking its camera. However, it is simply the definition of the slope, which is the height one rises when driving or walking divided by the hundred meters base measured horizontaly. One rises all the same of 80 meters by 100 meters of horizontal, which forms all the same an angle of almost 40 degrees (38,7).
The distance covered on the slope is then close to 130 meters. The 45 ° slope is particular because the horizontal distance equalizes with the rise, i.e 100 meters for both. The distance covered (diagonal of the triangle called then “right-angled”), is approximately 140 meters (141,5).
Slopes and distance to the meter.
On the road, a slope of 12% means that one rises (or goes down) of 12
meters by 100 meters gone on the horizontal line. The real distance
rode on the slope (the road) is hardly larger (100,72) because the
angle is still small. The angle corresponding is almost 7 degrees. A
descent known as “standard” a plane follows is 3 degrees slope, that is
to say roughly a 5% slope. Less than in a car ? yes, because the plane
flies much more quickly and thus goes down more quickly despite a
smaller slope. Moreover, on the basis of 3.000 or 10.000 meters, it
will cross a height much more important
Practically speaking :
one can measure the height of hillock, a building
or a mountain if, by knowing his distance (OH), one measures the
angle under which one sees the top of it.
This has been used for long
1 / On can measure the angle that makes the direction OP targeting to the summit, compared to the horizontal base (OH), which is the length of the shade ; one can calculate the height PH using trigonométry as seen before.
2 /One does not know how to measure the angle. In our environment vicinity, le rays of the sun come from the same direction everywhere, at any hour of the day. On this spot, the lenght of any shade has an identical ratio comparing to it's height. Plowing a vertical stick, one can find the ratio by dividing the length of the stick by its height (PH). The height of the building is egal to the lenght of its shade, divided by the lengfht of the stick shade, then multiplied by the stick height (identical ratio).
Ultra short waves (GPS), the laser and satellites allow very precise measures.
and sine waves. Warying our
angle of slope since 7° (slope of 12%), with 30° (sine 0,5 - slope of
44%), then 38,7 degrees (sine 0,625 - slope of 80%), finally of 45° (sines
0,70 - slope of 100%), one can observe that the slope turns like the
reverse of the needles of an alarm clock. Let us observe as comparison
the big hand of an alarm clock: the vertical
distance traversed by its point is small when
nearby the top (both sides of
the 12) or nearby the bas (both sides
from the 6), and large close to the horizontal line.
For an identical angle of rotation, that is to say an equal time, the traversed vertical distance is much more important close to horizontal than of the vertical. If one accelerated the movement of the alarm clock by removing the regulator and that one can see the needle by its profile and not by its face, one would see only the vertical movement of its big hand and one would see there is a stop at the top and bas, then acceleration, then maximum speed close to horizontal and so on. A curb could be obtained making sliding a band of paper, while putting down the different positions of the needle. This curve is a sinusoid (see below). translation : faibles variations (small variations).
Lunar tides and alternating electrical
current. The moon turns
around the ground and makes a full rotation in 28 days (roughly). The
moon exerts a variable force of attraction on all that lives on
our earth, and acts on
the reproduction of the species, mammals mainly
(tides for all the liquids, human include) or
the growing of plants. Like
the point of the needle, it describes its circle
around the ground and its force of attraction make vary
the tide relating to the same rules. There is a phase or the
tide is low, then it starts to go up slowly,
accelerates goes up (or goes down) very quickly, then slows down to
stop again at high tide.
To simplify, the sailors divide a cycle into three periods of equal duration. First period, it starts to go up slowly. Second period, it goes up quickly. Third period, it slows down before immobilizing to set out again in opposite direction. It also follows a movement progressive of acceleration, deceleration and passage at null speed of sinusoidal type.
The alternating electrical current is a kind of electric tide. With the same sinusoidal law, it starts by arriving slowly of a terminal of the socket, increases its force very quickly, then slows down to stop and to set out again again, this time by the other terminal. Like the tide, it changes direction. It arrives and it from goes away. Not in 6 hours or a bit less for the tide, but in a fiftieth of second (sixtieth in USA), that is to say fifty times a second. Physics 3 "electricity" (bas of page).
Between the points A and B, there is a complete cycle of the variation which is said a “period”. For the tide, it is a little less than 6 hours, for the alternating electrical current, 50 times a second in general, 60 in USA.
The ellipse is the curve all planets and stars follow in the their race around another
one larger, whose retains them by his enormous gravity (attraction due to its mass); like the moon around
the earth, earth around the sun and so on.
An ellipse can be drawn by planting two drawing pins or points on two points (called focus) and by
fixing on each one the tip of a thin thread.
This thread will be cut definitely longer than the space reserved between the points. One tightens then the
thread by pushing a pencil against it while having it run along (one cannot make so only half of the ellipse and it
is necessary to take again on the other side). The shape and the size of the ellipse will thus depend on the spacing
between the points and the length of the thread, which summarizes a mathematical formula.
Let us return to the thread: its length is always the same one (constant) wherever is the pencil which tightens it. Conclusion, in any point of an ellipse, the addition of the two distances towards the two points is constant.
is the curve that any stone launched in the air in direction of a target makes.
Even the ball of a rifle follows such a curve and it is necessary to regulate the sight (the rise) according to the distance. If the
way is very short, one can't see the curb and one believe, wrongly, that it is a straight line. In fact it is impossible because gravity inevitably draws the
stone or the ball downwards; it thus goes up by the power of the throw, then falls down more and more vertically. It is difficult
building a parabola with a thread because it is necessary that this thread be fixed at a point (the
focus) but slide at the other end along a straight line called director.
In any point of a parabola, the distances between the focus (the point) and a line called director, remain equal.
The chain curves is the curve which any thread tended between two points makes under the effect of gravity ; washing line or powerline, nothing escapes from it. One cannot plot this curve of which the mathematical resolution is complex and inaccurate because one makes in fact numerous estimations. Thus do not think the sag the cable makes to cross the river or to cross the valley could be reduced by tightening it more, either those of the bridges, whose weight for each meter is enormeous. They are already under a very strong tension and with the cold and the weight of snow, ready to break in spite of appearances. You will thus tend "in straight line" and only seemingly, a resistant and light thread on a short distance. It is what the masons practise for tracing a horizontal line on a wall: one coats the wire with a coloured powder, one tightens it well and one pinch/release it on the wall so that it deposits its color onto.
The curve of the dog is so called by reference to four dogs which would keep chasing while leaving each one the corner of a square. At every moment, the race of the one of the dogs is perpendicular to that of the pursued dog because it is the shortest way to reach it. As the dogs run themselves all after, the direction of each dog changes without stop. Each curve describes a kind of spiral. Without studies, without calculation, the dog follows this complex curve which is the best to catch up with the other which must also avoid the other.
This curve is only interesting to underline our aptitude (like the dogs considered), to correct our trajectory automatically in order to maintain a distances minimum (because that corresponds to the minimum effort). This concept of minimum effort and thus of distance minimum always pushes us to take the direct straight line at any moment to reach a spot. of which the pressed lawns whereas a path skakes aside, the reversed fences, angers to have to make a great loop where one passed before straight (motorways, properties), conflicts which go to wounds or dead of man if paths were vital (seen in New Caledonia). Without speaking about the toads one collects to make them cross the motorway at the season of love, of crabs crushed everywhere in the Pacific Islands and besides. All the life is conditioned by that : lots of current roads were old ways of passage, coming from the paths of the bas of the ages, not only from the men but also of the animals. One says in Corsica that to trace the path in the mountain, one follows the goat (which combines shortest with easiest). In the car, you often have two routes, shortest and fastest.