Theoretical reconstruction of Galileo's two-bucket experiment

In the present work, we address the solution of a problem extracted from a historical context, in which Galileo supposedly conducted an experiment to measure the percussion force of a water jet. To this end, the conservation equations of fluid mechanics in unsteady state are employed in the theoretical reconstruction of the experiment. The experimental apparatus consists of a balance, in which a counterweight hangs on to one of its extremities, and two buckets, in the same vertical, hang on to the other extremity. The water jet issuing from an orifice in the bottom of the upper bucket strikes the lower bucket. The objective is to find the jet percussion force on the lower bucket. The result of the analysis revealed that the method proposed by Galileo for the calculation of the jet percussion force is incorrect. The analysis also revealed that the resultant force during the process is practically null, which would make Galileo's account of the major movements of the balance credible, despite his having not identified all the forces acting on the system.


Introduction
Galileo's Discourses is originally divided into four days − as published in the Leiden edition of 1638 −, to which were posthumously added another two days, all written in dialogic form in Two New Sciences. The Sixth Day was translated by Stillman a e-mail: sbistafa@usp.br 2 Drake as the Added Day: On the Force of Percussion [1], where the specific goal of the interlocutors Salviati, Sagredo and Aproíno is to understand and find a means of measuring the percussion force.
The first experiment about this force discussed by the trio begins when Aproíno narrates to Sagredo an experiment with two buckets conducted by the Academic (Galileo) to investigate the effect of the percussion force. In this experiment (see Fig. 1), the upper bucket is filled with water and has a hole in the bottom. At the beginning of the experiment the orifice is closed, and the balance is in equilibrium. Once the orifice is opened, the water flows to the lower bucket. Initially the balance tilts to the counterweight side, and after the jet hits the lower bucket the equilibrium is reestablished. The present study has the objective of obtaining the forces acting on the balance in unsteady state, since the opening of the orifice in the bottom of the upper bucket until the end of the process, when all the water contained in this bucket has drained to the lower bucket. In the development of the theoretical model, we shall use the conservation equations of fluid mechanics in unsteady state, in the so-called integral form: continuity, in the form of conservation of the volume flux; energy, in the form provided by Torricelli's law; and Newton's 2 nd law, best known in fluid mechanics as the linear momentum equation.

The flow through the orifice and the formation of the water jet
The volume flux = through the orifice is given by [ in which is the discharge coefficient, o S is the area of the orifice, ℎ = ℎ is the water height from the orifice up to the free surface of the water in the upper bucket at instant , and is the gravity. Since Torricelli's law says = 2 ℎ , then we can write Eq. 1 as = .
The discharge coefficient consists of the product of two other coefficients, namely: the contraction coefficient , and the velocity coefficient , such that = .
The origin of the contraction coefficient is due to the fact that, as experience shows, the liquid jet cross section at the plane of the orifice continues to contract, until reaching a minimum section, which occurs at a small distance from , called vena contracta, which is crossed by trajectories that are sensible straight and parallel, in which the velocity is uniform, and the pressure is atmospheric, with the contraction 4 coefficient theoretically given by = ≈ 0,611 1 . Torricelli's law refers to the velocity at the vena contracta: in the plane of the orifice, neither the pressure, nor the velocity are uniform, and the velocity is lower than the velocity at the vena contracta.
The velocity given by Torricelli's law = 2 ℎ is, however, a theoretical velocity that does not consider the fluid internal viscous forces. Thus, the actual velocity can be obtained by correcting the theoretical velocity with the velocity coefficient , whose value is experimentally obtained. In this way, the actual velocity at the vena contracta is , and given by = = 2 ℎ . From this, appears the expression for the volume flux through the orifice as Experience also shows that the cross section of the falling water jet continues to contract, assuming a tapered form as shown in Fig. 2. The shape of the jet during descent may be obtained by applying Bernoulli's equation between point , at elevation , and point , at elevation , in the form where is the velocity at the vena contracta, whose cross section has a radius $, is the velocity at the section whose radius is r(z), ρ is the density, and # and # are the absolute pressures at and , respectively.
Disregarding the surface tension effects on the shape of the jet, then Eq. 3 may be rewritten as where B = B = − is the jet height at instant .
The jet volume D , at instant , will be given by

Theoretical model for the two-bucket experiment in unsteady state
We present next, the linear momentum equation, in the form applicable to the unsteady flow in a control volume Ω, limited by the water contained in the bucket at each instant, as [3] H But, from continuity S R 2 ℎ , and then, Then, finally, for the upper bucket.
in which H ZY is the weight of the water contained in the upper bucket. J KI Y , as given by Eq. 10, is the resultant force acting on the water body contained in the upper bucket in unsteady state.

Resultant force on the lower bucket in unsteady state
As far as the lower bucket is concerned, the velocity of the jet as it strikes the bottom of this bucket changes its direction from axial to the radial direction; then, KI M = M −P I ' = 0. Therefore, for the lower bucket For a fixed control volume that incorporates the volume of the water jet, the continuity equation, in unsteady state, for an incompressible fluid may be written as in which = 2 ℎ is the volume flux through the orifice and D is the volume of the jet at each instant, as given by Eq. 7.
Applying Eq. 7 in the assessment of we have that On the other hand, the momentum flux of the lower bucket free surface will be given by * N KI = ! S R P I ' , where M is the free surface area of the lower bucket.
Then, from these results, we may write Eq. 8 for the lower bucket as J KI Y_ = H ZY_ P I ' + ! 3 ;1 + < = > 5 ? @ A / P I ' + ! S R P I ' , in which H ZY_ is the weight of the water contained in the lower bucket, with given by Eq. (13). J KI Y_ , as given by Eq. 14, is the resultant force acting on the water body contained in the lower bucket in unsteady state.

Resultant force on the balance in unsteady state
The resultant force on the balance J KI Y will be given by the sum of −J KI Y (Eq. 10) and −J KI Y_ (Eq. 14). By considering that the weight of the water   The following values for the flow coefficients were adopted in the calculations as representative of the process: = 0.63, = 0.97, = 0.61 [3].

Discussion
For Galileo, the percussion force would be equal to the weight of the jet that is suspended in the air between the waters in the two buckets [1]. However, Fig. 5 shows that the percussion force has a different behavior from the weight of the jet, with a value always greater during the upper bucket drainage.
Aproíno, in his talk [1], states that the weight of the jet would be 'certainly' 10 to 12 pounds, although Salviati indicates in his replica that there would be some uncertainty due to the 'difficulty in measuring the amount of the falling water'.
Although Aproíno does not mention at which instant of time this value would have been obtained, it may be admitted that it could be at the instant when the jet first strikes the 15 lower bucket. At this instant, the percussion force corresponds to, approximately, 2.8% of the weight of water contained in the system, which gives 1.75 pounds. 3 At this same instant of time, the weight of the jet corresponds, approximately, to 1.7% of the weight of water contained in the system, that is, 1.06 pounds, which is a much lower value than that estimated by Galileo.  approximately, in favor of the percussion force. This resultant will cause the balance to remain a little unbalanced toward the buckets side during the upper bucket drainage.

Conclusions
The analysis made demonstrates that the percussion force in the lower bucket does not correspond to the weight of the jet that is suspended in the air between the waters in the two buckets, upper and lower, as suggested by Galileo. In fact, the percussion force is proportional to the square of the jet velocity, assuming a value always greater than the weight of the jet during the upper bucket drainage.
During the upper bucket drainage, the balance will remain a little unbalanced toward the side of the buckets, but due to the small magnitude of resultant force, with a value practically constant, and around 6.4 grams only during the entire process -which would make the unbalance of the balance described by Galileo small enough to pass unnoticed -, indicates that the report of Galileo could be considered as being credible "[…] but the water had hardly begun to strike against the bottom of the lower bucket when the counterweight ceased to descend, and commenced to rise with very tranquil motion, restoring itself to equilibrium while water was still flowing, and upon reaching equilibrium it balanced and came to rest without passing a hairbreadth beyond." [1].