Computations on fall of the leaning tower with considering air resistance

The free fall of a sphere was studied by considering air buoyancy and resistance. After selecting the reasonable drag coefficient formula recommended by the literature, partial differential formulas on the motion of balls falling in the Leaning Tower of Pisa are solved. The variation process of acceleration, velocity and displacement over time during the falling process of two spheres is obtained. The research results indicate that the kinematics of free fall considering air resistance is different from that neglecting air resistance. Air Resistance must be considered in the free fall of the solid ball after 0.3005 seconds. In the free fall of the leaning tower, air resistance makes the solid ball land at 0.4418 seconds which is earlier than the hollow. The variation of the acceleration of a solid ball with time can be described by a second-order function a=-0.115t2 -0.022t+9.801. Velocity does not satisfy the product of acceleration and time but can be described as a polynomial function of the velocity variation with time v = c1t2 + c2t + c3 . About the relationship between displacement and time, for a solid sphere, displacement is proportional to 1.9526 power of time h=5.4945t1.9526 , while for a hollow sphere, it cannot be expressed by a function. The relationship between air resistance and velocity during the falling process of an iron ball satisfies a polynomial function Fd =c1v2 -c2v+c3 rather than a simple relationship where air resistance is directly proportional to the first or second power of speed, which was used in many papers as a reasonable assumption.


Introduction
Circular shape is the most widespread shape in nature and the choice of nature must have profound reasons.In fact, modern science and technology also prefer circular shapes.The natural falling motion of the ball or the flow over the ball and cylinder is the most extensive basic problem encountered in fluid mechanics, mechanical engineering, aerospace, aviation, military and other fields, such as the movement of river sediment, various settlements, and the migration of drilling cuttings, prediction and PIV measurement of atmospheric pollutant movement, as well as gas-solid two-phase flow such as pneumatic conveying, all of which require understanding the motion of a falling ball.Therefore, this is also the main reason for the continuous attention and extensive research on the fall motion of the ball or flow over the sphere over the past century.
In 1638, Galileo published a remarkable treatise, Dialogues Concerning Two New Sciences often referred to as the Discorsi [1].This work describes experiments on free-falling spheres, spheres rolling down an inclined plane and pendulums.The experiments convinced Galileo that, in the absence of any fluid resistance, bodies in free fall or on inclined planes move with (1) constant acceleration, (2) uniformly increasing velocity and (3) the distance traveled is proportional to the square of the time.
Today, we recognize these properties as the kinematic formulas for acceleration is a u const = =  , speed is u s at = =  and distance is s = at 2 /2.But when a ball falls in the air, it is impossible for it to move in free fall without air resistance.Free falling ignores the air resistance.During the falling process of a sphere, in addition to being subjected to gravity, it is also affected by air buoyancy and air resistance, with resistance playing a crucial role.The resistance is a function of the magnitude of velocity.There is no definite expression to quantitatively provide a functional relationship.Because of the magnitude of resistance, , where Cd is a function of Re and solid shape, satisfies different relationships with Re from small to large.With air resistance being considered, the kinematic formulas for acceleration is and distance is 2 2 s at = deviate from the real falling.Could such diversion be neglected in the free falling with air resistance?What will be the new function between acceleration, speed and distance over time?Some have studied the falling of the iron ball in the Leaning Tower of Pisa experiment [2][3][4][5][6][7][8], in which, the air resistance is often discussed in two cases that it is proportional to the speed or the square of the speed [2][3][4][5][6][7][8].The damped falling motion in two cases has been studied.The resistance is proportional to the speed.The magnitude of resistance is proportional to the square of velocity.Equivalent Lagrange and Hamiltonian functions are given for the two types of motion and different solving methods such as the first integration method, point transformation method, regularization transformation method and Hamilton Jacobi formula method applied for solving [8].
In fact, according to the calculation formula of air resistance, its value is not simply proportional to the speed or the square of the speed, because there is a non-linear relationship between the drag coefficient and the speed of the object.Therefore, it is inaccurate to simply divide the air resistance of the research object into the two situations mentioned above.In [6], the landing time of two balls in the Leaning Tower of Pisa experiment was calculated, but the results were slightly biased because the relationship between the drag coefficient and the speed of the object was also the first piecewise linear approximation.The drag coefficient satisfies the complex relationship between the drag coefficient Cd and the Reynolds number Re, which is usually measured through experiments.In [7], the Cd formula to compute air resistance was selected as Formula (1).By using MATLAB software, the two problems of falling in the Leaning Tower of Pisa were analyzed.However, the article did not explain the rationality of the selection and application of the Drag coefficient.
In a 2002 review, Hahn states [9]: 'Given the relatively small velocities, distances, and times, Galileo could safely assume that air resistance would not play a significant role'.Later in [9], Hahn states 'we will assume that air resistance was negligible...'.Neither of these assumptions is backed up with quantitative arguments.Galileo can be credited as the first to make the assumption about air resistance.In Discorsi, page 276, he argues that because sphere sizes, distances and speeds are small in projectile experiments, external and incidental resistances among which that of the medium is the most considerable, are scarcely observable [10].
This paper aims to resolve the partial differential formulas on the falling in the Leaning Tower of Pisa experiment with air resistance considered.It starts from the force analysis of the falling sphere and considers the influence of air resistance on the motion.This paper uses Excel to compile a calculation program to resolve the partial differential formulas for the leaning fall of two iron balls, and the kinematics of the falling within the large Re range 0<Re<2X10 5 are obtained.The laws of acceleration, velocity and displacement were solved and the results were displayed.And acceleration over time was never reported in the literature.

The basic formula of the motion of a sphere falling in the air
For a sphere with a mass of M, the general formula of motion in the air can be expressed as follows: where u is the instantaneous motion speed of the sphere, t is time and F   is the sum of the external forces experienced by the particles, which can be expressed by the following formula: F   ＝Gravity + buoyancy + drag + additional mass force + Basset force＋ Magnus force＋ Saffman force (3) The last two terms on the right side of the above formula are collectively referred to as lift, which is the resistance caused by the rotation of the object when moving in the medium.It is usually in small order and can be ignored in general.The other forces are calculated as follows, where a ρ , μ and ν are the density, dynamic viscosity and kinematic viscosity of the fluid respectively, g is the acceleration of gravity, d is the diameter of the sphere, s ρ is the density of sphere and u and a u are the vertical motion velocity of the sphere and the fluid respectively.
C is the drag coefficient.In the case of small Reynolds numbers ( Re<1 ), the Stokes resistance formula holds.The resistance term is expressed by using the following formula: The additional mass force is where relative acceleration is defined as m is the mass of a fluid with the same volume as a sphere, the additional mass force expresses the resistance of the fluid to obstruct the unsteady motion of the object and its effect is expressed as the increase of the apparent mass of the sphere, The force of the fluid on the sphere depends not only on the relative velocity of the sphere at that time (resistance takes Stokes formula) and the relative acceleration at that time (additional mass force), but also on the history of acceleration, this part is called the Basset force.
Comparing it to gravity, there are relative values.
Only in the early stage of accelerated motion, the Basset force is about 10% of gravity.Then the Basset force is important, otherwise, it can be ignored [9].For example, for the particles, 0.1 d mm = .The falling velocity of a single particle 0.54 / u cm s = in water is taken with T = 15°C.Then the Basset force can be ignored when The first term on the right side of the formula is the effective weight of the sphere in the air after considering buoyancy.The second item considers the resistance exerted by the fluid on the sphere.The third item is the added mass force that takes into account the acceleration change after the rising reflux.

Description of air resistance and Drag coefficient
Historians have either ignored air resistance or explicitly considered it to be negligible in folio 116 v and Galileo's other inclined plane experiments [10].This would appear to be a reasonable assumption.Air resistance forces are proportional to ρ˜ = ρ f /ρ s , the ratio of fluid and sphere densities.It does not harm to verify the assumption by simulating an experiment with and without air resistance.If air resistance is indeed negligible, practically no difference between simulations of motion in vacuum vs air will be found.
Drag coefficient Cd is a quantity related to fluid Reynolds number and Reynolds number is a quantity with dimension 1 [11].Among them, ρ is the fluid density, d is the characteristic length and μ is the viscosity of the liquid.The relationship between drag coefficient Cd and Reynolds number Re is complex which is usually measured through experiments.It can be seen that the relationship between air resistance and Reynolds number is not linear.
The drag coefficient can also be calculated according to the formula.The equation [12] compares and analyzes a large number of existing relationships between drag coefficient and Re, which recommends a formula for drag coefficient that is highly consistent with the experiments.Figure 1 shows the calculated Re and drag coefficient in this paper, which shows that it is reasonable to use the formula recommended in the literature to calculate the drag coefficient.shows the relationship between the drag coefficient Cd and the speed.It can be seen that the relationship between the drag coefficient and the speed is neither a linear proportional relationship nor a quadratic square relationship.
Figure 2 shows the relationship between air resistance and the velocity of a sphere.It can be seen from the figure that the relationship between air resistance and velocity follows a polynomial function Fd = c1v 2 -c2v+c3, rather than a simple relationship, namely, air drag is not proportional to the first or second power relationship of velocity.Therefore, it would be unreasonable to assume that air resistance is proportional to the first or second power of velocity.

Research on the falling of the Leaning Tower of Pisa under 0 < Re < 2X10 5 by Excel computations
The basic formula for the Excel program, which is designed to resolve the kinematics of fall balls, is still Formulas (11) and (13).The gravity, buoyancy and air resistance of the sphere are considered in this calculation.The fluid medium is air and the material of the sphere is iron.The detailed method to resolve Formula (13) in Excel is not open here.

Experimental verification of calculation method by Excel in this paper
In this paper, Excel is used to develop a calculation method considering the influence of air buoyancy and resistance on free fall, which can calculate the kinematics parameters of free fall of a sphere.In order to verify the calculation, this paper has carried out the same calculation as the experimental conditions of the free fall measurement of a small ball in the experiment [13].The results show that the experiment results are consistent with the calculation in this paper and the theoretical formula results in [1].
In [13], a paper tape made by free fall of a heavy hammer in the laboratory was used to continuously select 14 counting points starting from the 4th point and measured the distance from each counting point to the starting point.The figure shows the theoretical formula for the drop of a small ball, the computation results in this article and the experimental data in [13].It can be seen that the three curves are consistent, though there are two obvious data errors in the experiment.The formula for the drop of a small ball derived from literature is Formula (14).

Computations on the falling iron balls in the Leaning Tower of Pisa Experiment
The movement of objects falling has attracted people's attention since ancient times [10].Aristotle, an ancient Greek scholar, intuitively believed from perceptual experience that the speed of falling objects was related to their weight.The heavier the object is, the faster it falls.For example, fruits and dead leaves fall from trees.Fruits fall faster than dead leaves, Therefore, Aristotle pointed out in Chapter 8 of Volume 4 of his book "Physics" that "there are two reasons why we see a known heavy object falling faster than another light object: either due to in different media passing through (such as in water, soil, or air), or other similar situations, but only due to the different weights of various moving objects."However, the thought of Aristotle, who ruled people for nearly 2000 years, was doubted by Galileo, the father of modern science.He refuted this conclusion with strict logic and reached the scientific conclusion that all objects fall vertically at the same acceleration.It is said that he also climbed the Leaning Tower of Pisa 54.5 m high one day in 1590, letting two iron balls of different sizes (one of which is 10 times heavier than the other) start falling at the same time, resulting in the almost simultaneous landing of both balls, thus negating the erroneous conclusion that "heavy objects fall faster than light objects".The Leaning Tower of Pisa experiment is a typical experiment to study the motion of free fall.Through this experiment, we can have the conclusion that two balls with the same volume and different masses will land at the same time when the influence of air resistance is ignored.But from another perspective, if the influence of air resistance and other factors is considered, what are the rules of the motion of the two balls considering air resistance?In [6], given that the height of Leaning Tower of Pisa h = 54.5 meters, ρ iron = 7. 9 × 10 3 kg/•m 3 with a radius of r = 0.02 m for both iron balls.However, the mass of a solid ball is 10 times that of a hollow ball.These parameters are used in the computations of this paper.

Results of computations and discussion
Table 1 gives the kinematics parameters of the two balls landing on the ground.We can compare the difference between the motion of a solid ball and a hollow ball.It can be seen from the table that the solid ball lands 0.4418 seconds earlier than the hollow ball, while air resistance is considered and the landing speed difference between the two is 7.639 m/s.
Through the calculation in this paper, the nonlinear motion of the falling solid iron ball (ball 1) and hollow ball (ball 2) of the Leaning Tower of Pisa is described.The falling process is calculated.At first, their accelerations are all gravitational acceleration and only gravity.As the falling produces air resistance, the fluid begins to accelerate the sphere, making the total acceleration gradually reduce from gravitational acceleration to zero, then the sphere starts to move at a uniform speed.From Figure 4, it can be seen that during the falling of two iron balls, their acceleration is similar with no drag motion in the initial time of falling and the influence of nonlinearity is not significant.However, after a short period of time, at 0.3005 seconds, the difference in density leads to different motions.In the early stages of motion, the rate of change in acceleration of a hollow iron ball is greater than that of a solid ball.The change of acceleration of a solid ball with time can be described by the polynomial function a=-0.115t 2 -0.022t+9.8017.However, if the falling height continues to increase, which is greater than the height of the Leaning Tower of Pisa 54.5 m, the acceleration cannot be directly described by the polynomial function.However, the acceleration of the whole falling movement must be divided into sections and described separately in stages which can fully express the acceleration of the movement by function.The polynomial approximation of the hollow sphere is shown in the figure .From Figure 3, it can be seen that the acceleration of the falling hollow sphere cannot be directly described as a function.Figure 4 shows the acceleration over time of different spheres falling in [1].It can be seen that the acceleration of two iron spheres falling is only one part of the falling process in Figure 5. Figure 6 shows the variation of the falling velocity of two iron balls over time, while Figure 7 shows the velocity variation of the falling ball in [1].From the figure, it can be seen that the speed in Figure 6 is only one part of Figure 7.Both iron balls do not reach uniform speed before landing on the ground.Figure 6 illustrates the falling process of two iron balls.During the initial fall, the velocity is similar with no drag falling.After 0.3005s, the falling of two iron balls is different from each other, but both can be expressed as polynomial functions to describe the changes in velocity over time, v = c 1 t 2 + c 2 t +c 3 .Moreover, the hollow sphere changes quickly to uniform velocity, resulting in less total time for the three stages of motion from nonlinearity to uniform velocity.
The calculation of the falling displacement h in this article is generated by integration.Figure 8 shows the relationship between h and t during the falling process.As can be seen from the figure, the higher the density of the sphere is, the closer the relationship between h and t is to no drag falling.The smaller the density of the sphere is, the more the relationship between h and t deviates from the nodrag situation.In the initial time of the just falling, the movement displacement and the no drag movement coincide.After 0.3005s for the solid ball and 0.5377s for the hollow ball, the displacement of the two iron balls is different from the situation without resistance.The change of displacement over time of a solid ball can be described by a power function, h = 5.4945t 1.9526 .However, the density of the hollow sphere is low and the air resistance has a great impact, which cannot be expressed by the power function directly.Figure 9 shows the relationship between speed v and acceleration a during the falling.It can be seen from the figure that the relation between the velocity and acceleration of two spheres cannot be described by polynomial functions directly.When speed v is smaller than 2.93 m/s, the falling acceleration could be constant g. Figure 10 shows the relationship between h and u during the falling process.It can be drawn from the computation that when the falling speed of the ball is more than 2.93 m/s, under the influence of air resistance, the falling displacement of the two iron balls is different from each other.The two curves look like parabolas, but they are not indeed.Moreover, the relationship between h and v of the two iron balls cannot be expressed by functions directly.

Conclusions
The research results indicate that: Kinematics of free fall considering air resistance is different from that neglecting air resistance.Air resistance must be considered in free fall.In the free fall of the leaning tower, air resistance makes the solid ball land 0.4418 seconds earlier than the hollow.
During the falling under the condition of the Leaning Tower of Pisa, the change rate of acceleration of the iron ball with a small mass is greater than that of the solid ball with a large mass.The acceleration variation over time of a solid ball can be described by a second-order polynomial function, a=-0.115t 2 -0.022t+9.801.However, if the falling height continues to increase, which is greater than the height of the Leaning Tower of Pisa by 54.5 m, then the acceleration cannot be directly described by the polynomial function which must be divided into sections for the whole falling movement and described separately in stages that can fully express the acceleration of the movement by function.After falling 0.3005 seconds, air resistance must be considered for the solid ball.Velocity can be described as a function of the velocity variation with time u = c 1 t 2 + c 2 t + c 3 .
About the relationship between displacement and time, for a solid sphere, displacement is proportional to 1.9526 power of time h = 5.4945t 1.9526 , while for a hollow sphere, it cannot be expressed by a function directly.
The relationship between air resistance and velocity during the falling process of an iron ball satisfies a function Fd=c1v 2 -c2v+c3 rather than a simple relationship where air resistance is directly proportional to the first or second power of velocity.The relationship between air resistance and velocity during the falling satisfies a polynomial function rather than a simple relationship where air resistance is directly proportional to the first or second power of velocity. d 0 0

Figure 1 .
Figure 1.Drag coefficient applied in the process in this paper calculation.Figure 2. The relationship between air resistance and speed in this paper speed of a falling iron ball.

Figure 2 .
Figure 1.Drag coefficient applied in the process in this paper calculation.Figure 2. The relationship between air resistance and speed in this paper speed of a falling iron ball.
Figure1shows the calculated Re and drag coefficient in this paper, which shows that it is reasonable to use the formula recommended in the literature to calculate the drag coefficient.Figure2

Figure 4 .
Figure 4. Acceleration of Two Iron Balls Falling Over Time.

Figure 5 .
Figure 5. Acceleration changes during the falling process of a sphere in [1].

Figure. 6 Figure. 7
Figure.6 Speed Changes over time of two iron balls.Figure.7 Velocity variation of the falling sphere in [1].

Figure 8 .Figure 9 .
Figure 8. Displacement variation of the falling balls over time.

Figure. 10
Figure. 10 Relationship between displacement and velocity in the falling of two iron balls.

Table 1
Time and Speed of Two Iron Balls Landing at the Leaning Tower of Pisa.