Freefall (kinematic)

Recall of the demonstration

One considers a body of mass

m

subjected to the field of terrestrial Pesanteur

\ vec {G}

One writes the Basic principle of the Dynamique of Newton:

$m \ vec {has} =m \ vec {G}$ where $\ vec {has}$ is the Accélération body. Thus

$\ vec {has} = \ vec {G}$ (m is simplified: noted and affirmed by Galileo).

One integrates once, compared to time:

v = g.t + cste

where

v

is the vertical Speed.

If initial speed is null then $v = g.t$ .

This law is stated by Galileo, to the great displeasure of those which think that if initial speed is null, the mobile cannot advance!

One integrates once again compared to time: $z (T) = \ frac {1} {2} G t^2 + Z (0)$ .

One thus obtains the movement of a body in freefall.

Numerical application with recall of the units

One can notice that:

$m \ vec {G}$ is a force, the force of Pesanteur or weight in newton (NR). Thus $g$ is in N/kg, i.e. in m/ $s^2$ .
Its value on the surface of the ground is 9,81 N/kg with 45° of latitude. Thus, we would have the following data with, to make simple, 10 N/kg:

Following a question of Beeckman to Descartes (before 1629), if the mobile goes down from H in the time $t_1$ , time $t_2$ which it will put to still descend from H is: $t_2= (\ sqrt {2} - 1) t_1$

a traditional problem of the 17th century: that is to say the Egyptian Triangle right-angled AOB out of O; Vertical OA =3a and OB = 4a horizontal: a M1 mass slips without friction on course AOB (reamed out of O), the other mass m2 slips without friction according to AB: which arrives the first? (Answer: if OA is larger, m2 is at the head able] (see opposite (?) and Cycloid)

And Galileo?

Time of Galileo, one knew neither the derivative, neither the theory of Newton, nor the units. It is right after him that one effleure the problem. Just as one does not divide a vector by a vector, it was idiotic, at the time, to divide a distance by a time.

It is in fact satisfied to say that speed increased like 0+1, 1+2, 2+3, 3+4, etc, i.e. 1,3,5,7,… in equal times: this wants to say v = g.t in our language.
To affirm that it noted it in experiments is " rather douteux" (cf hereafter Experimentation).
However, it was an excellent theorist, he had been able to reason thus (but there is not trace of it):

f (t_1)

the speed reached at time T1 (or its median value, it does not matter). Then one must have according to the principle of relativity:

f (t_1) + F (t_2-t_1) = F (t_2)

Let us take

t_2 = 2t_1

, i.e.

f (2t_1) = 2 F (t_1)

. And more generally

f (n.t) =n F (T)

Donc speed at time n.t is N time larger than speed at time T.

It undoubtedly reasoned with the mean velocity between two times, because it says 1+2, 2+3, etc, but the reasoning remains exact.
In fact it reasoned in term of traversed way: it is clear that

d_0 = 0+1, \; d_1 = 1+2, \; d_2 = 2+3

, and of course, via the sum of N first odd: 1+ 3 + 5 +… = N ² (reasoning known for a very long time), it found Z proportional to N ².

To carry out an experiment, still it was necessary that it had a watch which gives rate. Dugas (history of mechanics) indicates that it is with a Clepsydre (for the case of the tilted plan) that it could make this measurement; it weighed a water mass running with constant flow; previously it used the rate of the song (his/her father was a famous musician).

Note:: it could also have made this experimentation: that is to say T1; t2 =t1+t and T3 =t1+2t, three consecutive times, where z1 is marked, z2, z3. Then, one can check 2.z2 - (z1 + z3) = constant independent of T1 and always being worth g.t^2 because 2. (t1+t) ^2 - + (t1+2t) ^2 = 2 .t^2. One can show that this is characteristic of Z ~ t^2.

At all events, concept of " force" of gravity is not clear. At Torricelli which handles the infinitesimal calculus best, certain expressions are very modern.

The Tower of Pisa

Never a ball of feather and a lead bullet do not fall according to the same law from fall and this because of resistance of the air.

Galileo knows it well, since he discusses it in his " of Motu" , written in Pisa about 1591. It is in the last years in Pisa that it is detached from the Scholastic, but it is only about 1604, with Padoue, that it has famous Z ~ t^2, via a double error of reasoning (letter with Paolo Sarpi).

And there no was experimental checking made since the Tower: it is a legend, a little like apple of Newton: to see the famous article of Koyré which ridicules this legend (Koyré, " Annals of the University of Paris" , 1937; article reproduced in : Koyré, " Studies of history of the thought scientifique" , PUF, 1966).

For more precise details, to see Fall with resistance of the air.

Realistic experimentation

One rather quickly shows that on a tilted level of angle has, only the component, according to the plan, of gravity intervenes: the fall is slowed down of a factor sin With (it is celebrates it Loi of the cords of the circle of Galileo , 1602).

While taking has sufficiently weak, for example, sin A = 1/100, then time to traverse 1 meter is of 4,47 S: it is possible to point z1, z2 and z3 each second, with enough of precision. But friction should be avoided; or to make roll a ball, without slip: it is shown that it is necessary to take account of a factor (1 (1+2/5)) = 5/7 (see Bearing on a tilted level). But with this near, one can check that the distance covered is well z~ t^2. Very astutely one joined up a tilted plan of angle B (by reaming the joint carefully); and Galileo and Torricelli had understood that the ball went back to the same height and that times were like carrée root; (1 / sin B).

Moreover Galileo had quickly understood that when a pendulum oscillated, which occurred on the left was identical to what occurred on the right. And he rightly interpreted it like a movement on a succession of tilted plans (cf cycloidal Pendule and simple Pendule). The conclusion was then obvious: the period T was proportional to sqrt (L). But it did not state it immediately!

On the other hand, T = 2 $\ pi \ sqrt (\ frac {L} {G})$ will come later with Torricelli and Huygens.

Following the analysis criticizes of Koyré (cf " Studies galiléennes"), one began to doubt the reality of the experiments of Galileo, holding the fate of gedanken experiments to them. One needed the intervention of Stillman Drake (1979) (included in its book, " Galileo: pioneer scientist" , University off Toronto Near, 1990, ISBN 0 8020 8725 3), so that the notes of practical works of Galileo are found.

Recent TP

The pupils drop a graduated vertical roll very heavy. Electric eyes indicate the passage of the notches has, B, C, D.

Or three photographs are taken with times T1, t2 = T1 + T and T3 = T1 + 2t. One makes calculations 2.z2 - (z1+z3) for the 4 points has, B, C, D and one takes the average: it is g.t^2. One varies T1: nothing changes (or almost). One varies T, and one carries g.t^2 according to t^2. The value of G, one suspects it, is approached by defect: there remains always the resistance of the air!

The pupils are disappointed, except if one makes an extrapolated evaluation when T decreases, then when T1 decreases: admittedly their value is not exact, but… the " must" is to point out that the values of G decrease rather systematically if T1 increases. And that at the bottom, in the course amount of time 2t, acceleration was G - av^2 = G B.t1^2. They precipitate again on their results. Large joy and large spite: it is much better, but nevertheless the dispersion of the results of a class is undeniable: one will not arrive at 4 ChS (Significant figures). The Pendule of Kater will remain unequalled with 5 CHS, during more than one century (cf Chute with resistance of the air).

Method of Sakuma (1970, with the BIPM)

Very pretty handling, rightly rewarded: one makes fall, in the vacuum, the mirror of an interferometer of Michelson, and one records the run of the fringes. Of course, small technological easy way, the mirror is a corner of cube, it is to say that it does not matter its orientation. One tested catapult-launching upwards and redescente, but finally, it is not so better.

The current precision is of 5 microgals (1 Gall = 10^ (- 5) N/kg): that is to say a precision of 10^ (- 12). One sees perfectly the lunisolar Force of tide (100 microgals) which is a component of the weight, variable, often neglected (cf Pesanteur): for example.

Precautions to be taken!

As soon as one seeks this precision with 12 ChS, it is necessary to pay great attention so that one writes: G, with this accuracy of 12 ChS, varies quickly with altitude H of measurement; each variation of R/2. 10 (- 12) = 3.2 micrometers is seen!

Therefore, actually, an intermediate value of G is measured: one must apply the formulas of Kepler, which is impressive! That is to say G out of O and G. (1+h/R) ^2 at the point has of fall: which is the intermediate point B where the experiment evaluates G?

Newton dealt with this problem: Either TO = R = 2a and MT = has (1+cos $\ theta$ ), the rigorous drop time, since the perigee O is t/T = $\ theta + sin \ theta$ ; what leads to the exact result. By approximation, one finds the point B such as OB = OA/6, with better than 10^ (- 12) near.

With the plate of Calern (OCA, Observatory of the Riviera), one sees the annual dilation of the plate of Calern (the correction of Bouguer remains the same one, but not that of Faye!) : it is to say the sophistication of measurements since Galileo.

Lastly, it is necessary to take care not to approach over or below too important masses. A contrario , to do it makes it possible to measure the constant of gravitation G (cf Expérience of Cavendish).

See too

Freefall
Fall with resistance of the air
Bearing on a tilted level
potential Energy (gravitational)

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