On the ball on a string demonstration of angular momentum conservation

The ‘ball on a string’ demonstration is a common tool used in physics education to illustrate the concept of conservation of angular momentum. However, various confounding factors can cause significant deviations from the idealized case, particularly under extreme conditions or when using low-stiffness pivots or high coefficients of friction. These factors include air resistance, contact friction at the pivot point, the mass of the ball and string, the angle of the string due to gravity, and the wobbling of the pivot point due to the centrifugal forces acting on it. In this work, we critically review by means of accurate simulations the adequateness of the ‘ball on a string’ demonstration in view of these confounding factors and provide recommendations for instructors on how to maximize the educational value of the demonstration while minimizing potential confusion for students. Our analysis suggests that a stiff pivot and avoiding extreme conditions are key to obtaining results that are in good agreement with the idealized case. We also caution instructors against using the demonstration without at least mentioning the confounding factors, as this may lead to a questionable understanding of the underlying physics principles.


Introduction
Instructors of physics are quite familiar with the role that approximations, simplifications, and idealizations play in physics education [1][2][3][4][5][6].These techniques help to simplify complex systems and make them more accessible to students, allowing them to solve problems that would otherwise be beyond their mathematical abilities.However, it is important to consider the extent to which these idealizations are realistic and how real-world systems may differ from these idealizations.The use of idealizations is common in basic physics, such as ignoring friction, air resistance, and the mass of strings and springs, and considering most objects as 'point particles' [3].While introductory physics courses often include a laboratory component, the level of analysis required to account for complicating factors may be limited [7].In fact, learning to identify and employ such idealizations is arguably an important part of learning how to 'think like a physicist' [3,[7][8][9].
Conservation of angular momentum (CoAM) is one of the pillars of classical mechanics and builds along with conservation of energy and of linear momentum the framework of conservation laws and thus of variational principles.As it is often the case for many topics in introductory physics, the effect of this fundamental law is qualitatively shown in classrooms by means of simple demonstrations [2,[4][5][6][7]9, 10] like a person spinning on a turntable with weights in their hands and changing their rotational speed by means of stretching and pulling in their arms.The vectorial nature of the law is usually shown instead by means of holding a rapidly-spinning bicycle wheel while sitting on a rotating chair and changing the rotation axis of the wheel.Another common, though perhaps less widespread, demonstration is the ball on a string (BoaS) which consists of spinning a small mass bound with a taut string around a pivot point and reducing the length of the string to induce an increase in the spinning speed of the mass [11] as shown in figure 1 [12].In this work, we critically review the adequateness of this demonstration in view of the numerous confounding factors that can cause, under realistic conditions, a significant deviation of its quantitative behavior compared to the simple idealized case.
The typical treatment of the BoaS in an undergraduate textbook is apt to allow numerous simplifications and idealizations in order to make the solution possible for first-year students.These idealizations include not only ignoring obvious real-world losses due to friction and drag, but simplifying various features of the physical system.A real-world ball on a string departs from its textbook idealization in the following ways: 1.The ball and string will experience air resistance, which increases as the speed increases; 2. The string experiences contact friction at the pivot point, which will also increase with speed as a result of the growing centrifugal pull; 3. The ball is not a point mass, and its physical moment of inertia may be important when the string's length is small; 4. The string is not massless (though this is likely a very small factor); 5.The string is not perfectly horizontal, due to the downward force of gravity, and the angle of the string changes as its length decreases, meaning that the actual length of the string and the radius of the circle are not quite the same.6.The central support is not infinitely rigid and can 'wobble' when the ball is moving fast, meaning that the ball is not an isolated object and not the only part of the system with angular momentum.
The various sources of loss are all ignored in the traditional introductory textbook treatments [12], which raises the question of how good the various approximations are, and under what conditions they hold.For example, if we start with a golf ball on a 1 m string rotating at 10 m s −1 (i.e. less than 2 rps) and reduce the radius to 10 cm we find that the final speed of the ball predicted by the idealized CoAM treatment is 100 m s −1 which corresponds to a spinning rate of 9500 rpm.Such unreasonably large speeds predicted by reducing the radius to very small values has even led one internet commenter to doubt the validity of analyzing any system in terms of CoAM entirely [13].Leaving aside the obvious incorrectness of this conclusion, one shall not underestimate the resulting obvious finding that the demonstration could bring about such an extreme level of confusion in a novice to the point of making them questioning the validity of one of the pillars of classical mechanics.It is thus natural to question its pedagogical effectiveness at least to some degree.In the following, we provide insights as to why the casual use of this demonstration could be confusing and even detrimental for beginners unless care is taken to introduce the various sources of error and their relative importance.Our aim is to provide a more nuanced view of the BoaS demonstration, highlighting both its pedagogical value and limitations.

Methods
We consider a ball of mass m and radius r m spinning with speed v at a distance r from a pivot point.The tether is pulled in at constant speed u from the initial distance r 1 to the final one r .Correspondingly the speed changes from the initial value v 1 to the new, to be determined level v .
2 The following assumptions are applied: 1.The pull on the tether results in a Coriolis-like force acting on the ball: / F m uv r.Cor = 2.A friction force acts at a distance r p from the pivot.This is modeled with a friction coefficient m and considering only the centripetal force: / F mv r.The resulting model is schematically represented in figure 2. This final assumption results in a variable displacement of the mass m a a by an amount r a given by the centripetal force F : The effect of this displacement on the ball can be modeled considering the loss of angular momentum in favor of the mass m .
a It can be easily shown that the resulting Coriolis force on the ball gets modified such that an additional term emerges (see below for the detailed calculation):

= +
We can factor in all these forces in a discrete-time simulation to obtain the time dependence of the resulting speed v by means of numerically integrating the resulting differential equation for which no trivial analytic solution is available: We can thus in particular obtain the final value v 2 as shown in the 'Results' section.

Model for non-fixed pivot
The calculation of the modified Coriolis force is performed by means of several approximations assumed to obtain an iterative picture that could be used as a model for beginners, although a somewhat difficult one.We determine the dragging effect of the mass m a on the ball in absence on other forces and transfer the result to the general case.This is not perfectly accurate, but it makes sense within the general idea of considering the individual dissipative contributions independently and the exact treatment is provided in section 2.2 anyway.Additionally, we neglect the finite size of the ball regarding its moment of inertia and we only consider its effect on the centrifugal force neglecting that of the mass m .a Finally, we recall that r is the distance between the masses (s.figure 2) but we assume that r is much larger than r a at any moment so that r r a + can be replaced with r.Since we are only dealing with the interplay between the two bodies, these approximations allow a computation based on instantaneous CoAM between the ball of mass m at distance r and the pivotal mass m a at distance r : The first term in equation ( 10) is the Coriolis-like force acting on the ball and the second term is the 'drag' induced by the presence of the mass m a at distance r .a We can thus finally write down the corresponding 'effective' force caused by it:

Full wobble treatment
A complete treatment of the wobble-effect requires a combination of all forces as the presence of the additional inertia represented by the mass m a alters in principle the effects of friction and air-drag as well.The only effects that are still neglected in this advanced treatment are gravity and the string (assumed inextensible, massless, and of zero size).The disadvantage of this advanced analysis is that the various contributions get tangled together and it becomes impossible to accurately analyze their contribution individually.Moreover, such a complex treatment is mathematically more advanced and thus even less suitable for beginners.The best approach to this analysis is to rely on energetic considerations, namely we equate the change in energy E with the power P produced by the various forces in action: ( ) We can first write down the total energy of the system: with its initial value: where we assumed that r a is initially equal 0. For the power, the following forces must be considered: the centripetal force F m c acting on the mass m, that F ca acting on the mass m , a the friction force F , fr and the drag force F dr ( )  Representing the central pivot as a hollow tube, its elastic constant k is mainly given by the bending of the tube itself.This can be modeled with the Euler-Bernoulli equation [14]: where t is the applied torque, E is the Young's modulus of the tube material, I z x y d d 2 ò = is the second moment of the area [15], and ( ) w x is the deformation of the beam as a function of the position x along its length.For a tube of inner and outer radius r 1 and r 2 the area moment reads for the beam's deformation, the displacement w D at the free end satisfies: with l length of the tube and this equates to where we made use of the fact that F l, t = assuming that the force F acts on the free end of the tube.Finally, writing the displacement in an elastic form we get an expression for the elastic constant: ´respectively.

Results and discussion
In the following, we define standardized values assumed for the relevant parameters unless otherwise stated.The ball has a mass of 0.050 kg and a radius of 0.025 m which corresponds to a typical golf ball.The pivot is assumed to be a tube of radius 0.002 mm with an elastic constant k 2000 N m .A stiffer pivot can be achieved by means of fixing the tether-guiding tube to a massive support instead of hand-holding it.
Friction at the central pivot is modelled with a coefficient 0.500 m = acting on a radius r 0.002 m. p = The drag coefficient C is set at the standard value for a sphere (C 0.47 = ) and air density is assumed to be 1.2 kg m .
3 r = -The parameters values are summarized in table 1 and are used to simulate several scenarios according to equation (2).These entail a basic simulation where the string is pulled from 1.0 m down to 0.5 m within 1.0 s (figure 3) and an 'extreme' one where the string is shortened down to 0.1 m within 1.0 s (figure 4).The latter has been also simulated for a 'stiff' pivot with k 1.4 10 N m 6 1

=
´as calculated above (figure 5) or with a 'fast' pull-in happening in 0.1 s (figure 6). Figure 7  = ´-).Finally, the basic simulation with full treatment of the wobble-effect according to equations (20) and (21) is shown in figure 8.In all simulations, the resulting angular velocity is compared with that predicted by a simple CoAM-only model for reference.
The simulation of figure 3 does not deviate significantly from the idealized scenario but it is also worth mentioning that the typical radius reduction that is used upon demonstrating the effect can easily be larger than just 50% and often resembles more the conditions of figure 4 (i.e.radius down to 10%).In this case, major losses are observed as the ball gets close to the pivot point.Down to a radius reduction of 60%, only 20% of the angular momentum is dissipated, compared to 90% angular momentum lost when the radius is reduced to 10% of the initial value.It is also apparent that losses are dominated by the wobble of the support while air-drag appears less significant.With k 2000 N m 1 = -(corresponding to an arm bending by 5 cm when holding a weight of 10 kg) we achieve a reasonable wobble for the handheld situation of up to 5 cm and the final rotation is around 1200 rpm.Assuming a stiff pivot (i.e. for very large values of k) as in figure 5, losses are less pronounced and dominated by friction.Dissipation can be further reduced by means of pulling in faster: for instance, with a pull-in time of 0.1 s like in figure 6, we can retain 90% of the original angular momentum even if the radius is reduced to 10% of the initial value.This is similar to what can be obtained upon simulating an ideal environment with very low friction 0.05 m = and near vacuum 0.012 kg m 1 r = -with a pull-in time of 1.0 s as in figure 7. It is worth noticing that the configuration that retains 90% of the angular momentum by means of a fast pull-in, while conceptually feasible, is likely almost impossible to realize in practice due to the resulting massive strain on the tether ( / F m v r 7100 N 2

= =
).This would either result in tether failure or a practical impossibility of acting the pull-in sufficiently fast as the required power approaches the value Fu 6400 W. =  The full-scale simulation of figure 8 is additionally able to catch vibrations at the pivot point that are indeed realistically expected.Nevertheless, the overall result is impressively similar to that obtained with the simpler model of figure 3, a finding that we could confirm on the other  ).simulations as well.It is also worth mentioning that, while formally more correct, this simulation strategy is more prone to numerical instabilities than the simpler approach of equation (2).

Conclusion
The 'ball on a string' demonstration is a common way for instructors to illustrate the concept of conservation of angular momentum in introductory physics courses.Our analysis has shown that there are several confounding factors that can significantly affect the behavior of this system, particularly under realistic conditions, e.g. when the reduction of the string is shortened to less than 50% of its initial length or the pull-in is not fast enough.One of the main take-home messages from this study is that the conditions under which the 'ball on a string' demonstration is performed must be carefully considered because the resulting deviations from the idealized case can be quite significant.For example, our simulations show that the effect of the wobbling of the pivot point can be quite pronounced when the elastic constant of the pivot is low and the pull-in time is slow.Similarly, the effect of friction at the pivot point becomes relevant when the coefficient of friction is high and the pull-in time is slow.Finally, air resistance also plays a role in the 'ball on a string' demonstration, although,  quite surprisingly, its effect is generally less pronounced than the other factors.There could be more subtle effects connected to the central mass lagging behind the ball that cannot be captured by the current model from figure 2 and would require an even more complex treatment with two degrees of freedom like in [11].We believe though that this level of analysis would be out of scope because it is certainly beyond the reach of introductory physics students and our work already shows satisfactorily how the simple lossless model can be arbitrarily off and in what order the confounding factors contribute to the discrepancies.Given these confounding factors, it is questionable to use the 'ball on a string' demonstration without at least mentioning them to students.Instructors should be aware of these factors and consider the extent to which the idealized case deviates from the real-world system.It is also worth noting that the hierarchy of the confounding factors can be somewhat counterintuitive.For example, our simulations show that the effect of the wobbling mass at the pivot point is generally more pronounced than the effect of friction or air-drag, even though these two are typically the first causes that come to mind when discussing the observed deviations from the ideal case.This highlights the importance of carefully considering the specific conditions of the demonstration, as the relative importance of the confounding factors can vary depending on the specific parameters of the system.
Our analysis suggests therefore that a stiff pivot is of paramount importance in the 'ball on a string' demonstration, as it minimizes the losses due to the wobbling mass.A fast pull-in also contributes significantly to reducing the deviations from the idealized case, particularly when combined with a low elastic constant for the pivot or a non-negligible friction coefficient.In the light of these findings, demonstrations and experiments that are based on symmetric mass distributions, as opposed to the 'ball on a string' demonstration, may be more appropriate for illustrating the concept of conservation of angular momentum, as they allow students to focus on the key physics principles without being distracted by the confounding factors that can arise in more complex systems like the one discussed here.It is also important to note that the 'ball on a string' demonstration is just one example of how approximations, simplifications, and idealizations are used in physics education.There are many other examples of idealized systems that are commonly used to illustrate key physics principles, and it is important for instructors to carefully consider the extent to which these idealizations are realistic and how real-world systems may differ from them.By doing so, instructors can help students develop a deeper understanding of the underlying physics principles and better appreciate the limitations of the idealized systems that are used to illustrate them.

2 p
ball too.Here C is the drag coefficient, r is air density, A r m = is the ball's cross surface.

Figure 1 .
Figure 1.Typical schematization of the ball on a string demonstration as a sample problem of an introductory physics book [12].

4 .
The wobbling of central pivot is modeled with a mass m a attached to the fixed center by a spring of constant k. 5.The central point, the mass m and the mass m a are on the same line, i.e. the motion has only one degree of freedom.

Figure 2 .
Figure 2. Schematic model of the ball on a string demonstration including a non-fixed pivot point.
For the complete solution we need though the time dependence of r a which we can extract from the corresponding equation of motion:

3 = 2 2 =
As examples of the typical resulting values, we consider a tube made of steel (E For tube length values l of 0.20 m and 0.02 m the resulting elastic constant amounts to 2700 N m 1 -and 1.4 10 N m , 6 1

1 =-
This value corresponds to the strength of an arm flexing by 0.05 m when holding a mass of 10 kg and it is comparable with the above estimate for the longer tube.The mass of the arm is set at m 2.0 kg. a = represents, as a term of comparison, an 'ideal' simulation with very low friction (

Figure 6 .
Figure 6.Extreme demonstration r 1.00 m 0.10 m, =  stiff, pull-in time reduced from 1.0 s to 0.1 s.

Table 1 .
Standardized values assumed for the simulations parameters unless otherwise stated.