The toppling iPad

The motion of a toppling iPad is analyzed in detail. Measurements of angular velocity and linear acceleration were made for toppling about the edges and corners. From the analysis of the angular velocity, the position of the center of mass and the gyration radii were determined. The position of the accelerometer in the iPad was measured using both the centrifugal acceleration and the Euler acceleration, with the latter measurement providing more accurate data. This experiment could be used in the first year of a university course in mechanics, either in a physics laboratory or as a home experiment.


Introduction
In experiments with tablets and smartphones, one of the simplest experiments is to measure the angular velocity and acceleration while the smartphone is toppled over from a nearly vertical position [1][2][3]. This experiment is easy to perform as it does not require any special accessories, and it is quite controlled. The analysis provides a lot of insight into concepts such as moment of inertia, gyration radii, pendulum motion at large amplitudes, and the importance of virtual accelerations, and it is complex enough to be appropriate for a first-year physics course.
Here, the toppling of an iPad around its four edges and four corners was measured. The analysis is described in detail and applied to the different data sets in order to obtain information about the location of the center of mass, the gyration radii, and the position of the accelerometer. The latter is often useful for accurate accelerometer measurements since it generally does not coincide with the center of mass. Several methods for localization have been proposed, all of which rely on attaching the smartphone to a rotating device and measuring the centrifugal acceleration [4][5][6]. The present data show that the position of the accelerometer is better determined by the Euler acceleration. The analysis also shows that there is a slight imbalance of the center of mass with respect to the different edges.

Experimental details and basic theory
An iPad Air 3 with dimensions a = 174.1 mm, b = 250.6 mm, and thickness c = 6.1 mm was used in the experiment. The iPad was chosen because the gyroscope, accelerometer, and linear accelerometer data are all measured at the same rate of 100 Hz. This allows a direct point-by-point comparison of the data without the need for any interpolation. The orientation of the xyz reference frame attached to the iPad was standard such that the x-axis is parallel to the side a, the y-axis is parallel to the side b, and the z-axis is perpendicular to the display. Two different experiments were performed. (1) Standing on an edge in a nearly upright position, the iPad was released and toppled about one of its four edges, stabilized by a heavy book (2.8 kg, 65 mm thick) to prevent it from sliding backwards on the surface. (2) Standing on a corner with the diagonal almost upright, the iPad was released and rotated around one of the corners toward one of the long edges. The latter movement was stabilized by two heavy books on both sides of the iPad, such that the rotation was mainly about the z-axis. Since this is supposed to be a home lab for students, the release mechanism was simply a finger lifted away from the iPad. The iPad fell onto a 2-3 mm thick stack of papers to prevent any damage. During the toppling, the angular velocity w  and the linear acceleration = + b a g    (acceleration without g), which measures only the inertial acceleration components, were recorded using the app 'phyphox' [7]. An image of the iPad is shown in figure 1, with a definition of the angle j. The moment of inertia about the axis of rotation-either the pivot line along an edge or an axis through the pivot point at a corner-is denoted by I, the minimum distance between the axis of rotation and the center of mass by s. With the respective gyration radius R and using the parallel axis theorem, the moment of inertia can be written as Friction affects the motion both through air drag and the slippage of the iPad at the pivot line or point. Within the accuracy of the measurements presented here, air drag can be neglected. Slippage, however, is inevitable even for large friction coefficients [8,9]. It is neglected here, since the toppling about the edges was stabilized by a book for backward slippage and since data affected by slipping in the forward direction was excluded from the analysis. In case of toppling about a corner slipping was not prevented, and the data showed to be more affected by slipping, but also wobbling than the data for toppling about the edges. These data were only used for consistency checks. In the approximation that slipping and air drag is neglected, the equation of motion is [10,11] ( ) j j = I mgs sin , 2 with the angular accelerationj, the iPad mass m and the acceleration due to gravity g = 9.81 m s −2 . Multiplying by the angular velocity j  and integrating from initial time t 0 = 0 to time t, or conservation of energy, leads to The motion starts with an initial angular velocity j 0  at an angle j 0 and with an initial energy E 0 . Equation (3) describes the large angle motion of a pendulum [11], and can be readily solved for the angular velocity with The differential equation can be solved and written as [10,11] ( ( ) ) ( ( ) ) ( ) with the incomplete integral of the first kind F(x, k) and 0 k 1. The latter equation is an implicit solution to this problem.
In the present experiments, the iPad starts from rest, j = 0 0  , with an energy actually exceeds unity. Since the domain of the elliptic integral can be extended to k > 1. Since the Jacobi elliptic function ( ) y k sn , is the inverse function to the elliptic integral y = F(x, k), equation (8) can be rewritten as Both formulations are important if the experiments are performed with nonzero initial angular velocities and/or with different starting angles.
In the case of toppling about the edges, to a good approximation, both the angular velocity j 0  and the initial angle j 0 were close to zero, such that k ; 1. In this case, the integrals in equation (8) can be readily integrated to give such that explicit solutions for the tilt angle j and the angular velocity j  are found: Up to this point, the analysis uses only the angle j, the angular velocity j  and the angular accelerationj. The angular velocity is measured with the gyroscope, the angle and the angular acceleration are determined by integration and differentiation, respectively.

Linear acceleration, toppling about an edge
Furthermore, the measured linear acceleration can be used to infer the position of the accelerometer in the iPad. Let us first discuss toppling about the edges. If the minimum distance between the sensor and the pivot edge (x-axis for illustration) is denoted by l, the Eur. J. Phys. 44 (2023) 035003 M Ziese centrifugal acceleration b y along the y-axis is and the Euler acceleration along the z-direction is given by with the angular velocity component ω x . This results in two ways to determine l as the slope of (1) the b y versus w x 2 and (2) the b z versus w x  curve. The first method is the conventional approach [4][5][6], and the second is an alternative method studied in this work. The latter is of the first order in angular acceleration and should be more sensitive for small values of angular velocity. In the case of toppling about the y-axis, the x and y indices must be swapped in the equations.

Linear acceleration, toppling about a corner
Toppling about the corners can also be used to determine the position of the accelerometer. In this case, l is defined as the distance between the sensor and the corner, and the centrifugal acceleration is given by l , 1 7 x z 2 and the Euler acceleration by Since the centrifugal acceleration is along the line connecting the corner to the sensor, and the Euler acceleration is perpendicular to this line, see the red coordinate system in figure 1, the white accelerometer coordinate system in figure 1 is not aligned with these directions. Accordingly, ¢ b x and ¢ b y are obtained from the measured linear acceleration components b x and b y by a rotation through the angle ò which is defined in figure 1: cos . 20 According to the basic theory presented in the previous section, information about the moment of inertia and the position of the center of mass can be obtained in several ways from the analysis of tilt angle j, angular velocity j  and angular accelerationj [2]. Here, the angle j was calculated from the measured angular velocity by numerical integration and plotted against time in figures 2(a) and (d). Fitting equation (13) to the data gave j 0 and ω 0i . The shoulders of the red and blue curves above the fitting line in figure 2(a) might indicate some slipping, see [8]. According to equation (4) the square of the angular velocity j 2  depends linearly on ( ) j cos 2 2 . Since the toppling starts from a nearly upright position, j 0 ; 0,ω 0i was obtained both from the intercept and slope of linear fits to the data shown in figures 2(b), (e). Within experimental accuracy, intercept and slope were found to be identical. Finally, the angular acceleration was calculated from the angular velocity by numerical differentiation and plotted against   Table 1. Angular frequency ω 0i and distance s i of the center of mass from the edges i = b and l with (b, t, l, r) = (bottom, top, left, right). Radius of gyration R i for rotation around an axis through the center of mass parallel to the x-(R x ) or the y-axis (R y ). The theoretical values in the last two columns were calculated from the dimensions under the assumption that the iPad is a uniform rectangular block.
Equation Equation From pairwise data for the angular frequencies (bottom and top, left and right) and the constraints s b + s t = b and s l + s r = a, values for s i and R i were calculated from [2]   curves seems more trustworthy. In any case, all data sets consistently indicate a 2-9 mm downward shift of the center of mass. Compared to the height of 250.6 mm this is small, so placing the center of mass in the middle of the instrument should be a good approximation for most measurements.

3.1.2.
Toppling about a corner. Several toppling runs were performed around the four corners, labeled bl, br, tl, and tr ('bottom left' and equivalent). In this case, the iPad diagonal was nearly vertical at the beginning, so a tilt of the long edge toward the horizontal corresponded to a rotation by an angle of ( ) p -=  a b 2 arctan 55.2 . The data were analyzed according to the procedure described in the previous section, i.e. the parameter ω 0 was determined by fitting to the j − t, ( ) j j cos 2 2 2  and̈( ) j j sin curves, see figure 3. Compared to the measurements of toppling about the edges, toppling about a corner is more affected by backward slippage, as the iPad cannot be stabilized easily against this [12]. Moreover, although the motion is guided, the iPad still wobbles in the gap between the books. Slippage can be seen in figure 3(d) in the strong deviations of the measured curves from the theory. Therefore, the present measurements are not sufficiently precise to determine differences in the distances of the CM from the corners.
The data are only used as a cross-check. The distance between the iPad center and the corner was 148 mm, slightly smaller than + = a b 2 152.6 mm 2 2 due to the rounding of the corners. The average ω 0 values of 6.8 ± 0.2 s −1 (j − t and ( ) j j cos 2 2 2  ) and 7.0 ± 0.2 s −1 (̈( ) j j sin ) resulted in radii of gyration about the z-axis of R z = (98 ± 5) mm and (90 ± 8) mm, see table 2. The uncertainty for these values was significantly larger than for the other directions, see table 1. This is attributed to two factors, namely the shift of the pivot point along the rounded edge during tilting and the slightly wobbly motion, see the curves in figures 3(c), (f). The theoretical value, calculated for a uniform rectangular block with sharp edges, is . This value agrees with the third experimental value within the uncertainty. Since c is much smaller than a and b, the perpendicular axis theorem  holds within the measurement uncertainties for two of the three methods, see table 2. The results of this and the previous section show that data evaluation using thë( ) j j sin curves gives the most accurate results.   Table 2. Angular frequency ω 0 and radius of gyration R z as determined from toppling around the corners.
Equation l b + l t = (202 ± 10) mm fall short of the manufacturer's specifications for dimensions a and b by 20% and 30%, respectively. Thus, the accelerometer position determined by this method has a systematic uncertainty of at least ±20 mm. In particular, for the toppling runs around the y-direction, see figures 4(c) and (d), the measured slopes are clearly larger than two. If a general power law b = lω p is fitted to the data, the exponents p are 2.08 ± 0.03 (x b and x t ) for rotation around the x-axis and 2.33 ± 0.02 (y l ) and 2.46 ± 0.03 (y r ) for rotation around the yaxis. The prefactors decrease even more compared to the fits with a quadratic dependence, but cannot be interpreted anymore as distances. The smaller the distance between accelerometer   and edge, the smaller the centrifugal acceleration and stronger the influence of measurement uncertainties and noise in the acceleration measurement. This, however, does not quantitatively explain the strong deviations from the quadratic dependence for rotations around y, since the distance of the accelerometer from top edge is the smallest. It is also not straightforward to invoke a more significant influence of slip for toppling around the long edge.
In the alternative approach, the Euler acceleration b z is plotted against the angular acceleration about the respective tilt axis, as shown in figure 5. The angular acceleration was obtained by numerical differentiation of the measured angular velocity. The resulting curves are nicely linear and linear regression yields the distances between accelerometer and edges as shown in table 3. The sums l l + l r = (177.0 ± 1.0) mm and l b + l t = (247.4 ± 1.9) mm agree with the iPad dimensions within 1.7% and 1.3%, respectively, viz. within three and two standard deviations. This is an excellent agreement, indicating that the accelerometer position was accurately determined with an uncertainty of about ±2 mm.

3.2.2.
Toppling about a corner. Linear acceleration and angular velocity were measured when toppling about the four corners, in all cases moving from an unstable equilibrium with the iPad diagonal in a nearly vertical direction towards the long edge in horizontal direction. As described in the previous section, see equations (17) to (20), the analysis of the acceleration data is quite complicated, since the position of the accelerometer not only enters into the distance l between the corner and the sensor, but also into the calculation of the angle ò. In principle, l and ò can be determined from the relations However, as said before, the data for toppling about the corners is less accurate than for toppling about the edges, and therefore, the data are used only as a consistency check for the previously determined sensor position. The accelerometer is shifted rightward and upward with respect to the center of the display by (x S , y S ) = (12 ± 2, 54 ± 2) mm. From this, the four angles  (( ) ( )) =   a x b y arctan 2 2 S S and the linear acceleration components ¢ b x and ¢ b y were calculated from the measured data using equations (19) and (20). The resulting curves are shown in figure 6 for toppling about the bottom right and top right corners. In certain ranges, the centrifugal acceleration ¢ b y was found to be quadratic in ω z and the Euler acceleration ¢ b x was linear in w z  , with slopes equal to the distance l ij with i, j = b, t, l, r. Fitting and averaging gave the values listed in table 3. As before, the values obtained from the centrifugal acceleration are systematically lower than the values obtained from the Euler acceleration. The values measured directly on the iPad with a ruler from the rounded corners to the pre-determined sensor position are given in the last column of table 3. Within the measurement uncertainty, the distances between the accelerometer and the br, tr, and tl corners agree well, only the geometric distance for the bl corner is about 5% larger than the value determined from the Euler acceleration. Overall, this can be considered to be a convincing consistency check.

Toppling about an edge with various initial angles
As a final crosscheck, data were taken for toppling about the bottom edge, starting the runs at a finite angle between 2 and 80 degrees. Figure 7 shows both the angular difference j x − j x0 , obtained by integrating the angular velocity, and the Euler acceleration b z . The single parameter ω 0 = 7.65 s −1 was obtained from a fit of equation (13) to the data for an initial angle near zero. The other solid lines in figure 7 were obtained using equations (10) and (16) and l b = 177.3 mm, as determined in the previous section.
Overall, the theoretical curves agree quite well with the data. For initial angles greater than 10 degrees and times less than 30 ms, the measured Euler acceleration drops below the expected plateau value. This is observed in both the acceleration and linear acceleration data and is likely a sensor artifact due to the abrupt change in acceleration value by more than 3 m s −2 . Note that the toppling accelerates strongly even with an initial angle of 2 degrees, since the angular acceleration is finite from the beginning.

Conclusions
The toppling of an iPad was studied in the approximation of a physical pendulum at large amplitudes; slipping and damping were neglected. Within this approximation, the data for toppling around edges and corners could be quantitatively understood. The evaluation of the measurements provides data on the position of the center of mass and accelerometer as well as on the radii of gyration. The data indicates that the center of mass is located near the center of the device, with a small downward shift towards the home button. The radii of gyration are consistent with theoretical values calculated from the iPad's dimensions assuming constant density. With respect to the center of the display, the accelerometer is (54 ± 2) mm up and (12 ± 2) mm to the right. This is consistent with previous data [13].
Overall, the results are proof of principle that a challenging home lab in the first or second year of University Physics might be constructed from this experiment, which goes beyond the analysis presented in [1]. The home lab is centered on an exploration of pendulum motion beyond the small amplitude approximation. Not all experimental facets presented here need to be covered, but as the core, the experiment should comprise toppling around one edge such that the respective radius of gyration and the distance of the accelerometer from that edge are determined. At present, we assign this experiment to second-year students in our physics program.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.