Measuring the moment-of-inertia of a rigid body using a swing-pendulum

A novel way to construct a compound pendulum is to suspend a distributed mass from a single pivot using light inextensible strings. Here we describe how such a compound ‘swing-pendulum’ can be used to infer the moment-of-inertia of a rigid body. Our approach is particularly suitable for contexts in which it is impractical to suspend the body from one of its internal points, and is illustrated using data sourced from student-led experiments on steel and aluminium rods.


Introduction
The compound (physical) pendulum is a key topic in introductory physics, and one that connects several core concepts, including simple harmonic motion, centre-of-mass, moment-of-inertia, and torque [1][2][3][4].Such a pendulum is also easy to construct: one simply suspends a rigid body of mass m through a horizontal axis z passing through any point O excluding the body's centre-of-mass C (see Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.figure 1(a)) [1,4].Indeed, if this is done, then the the body can be set to oscillate harmonically with time-period where g ≈ 9.81 ms −2 is the acceleration due to gravity, I z is the body's moment-of-inertia about O, and h is the distance from O to C (see figure 1(a)) [1,4].Crucially, since the parallel axis theorem may be used to express I z in terms of the body's moment-of-inertia I about the horizontal axis passing through its centre-of-mass C, viz [6,7] In both cases the centreof-mass C of the body is a distance h from O, where OC makes an angle θ with the vertical OP, and swings along an arc PC.Thus, the dynamics of each configuration are identical, with the weight of the body mg yielding a restoring force F = mg sin θ perpendicular to OC (see section 2).[Note: as indicated by the x-y-z reference axis, the z-axis passing through O is normal to the x-y plane containing PC].
Equation ( 1) can also be expressed as In this way, one may infer the body's moment-ofinertia I empirically by measuring the pendulum's time-period T for a given h [6,7].Almost all textbook analyses of the compound pendulum use a pivot O internal to the body (see figure 1(a)) [5], and it is therefore no surprise that experiments reported in the wider literature typically do the same, either by forcing a pivot through the body, or by boring a hole as a point of suspension [6][7][8][9][10][11].Such approaches are effective if the rigid body is relatively small, and made from a low-friction material that can be worked easily; however, if the body is impractically large or heavy, or if damage by boring is to be avoided, then fashioning an internal pivot is more difficult.
In this article, therefore, we describe a novel, alternative method for constructing a compound pendulum that we refer to as the compound swingpendulum, and which is based on swinging a rigid body from an external pivot using light inextensible strings (figure 1(b)).Crucially, since this 'swing-pendulum' approach does not require boring holes, or forcing pivots, it can be used to infer the moment-of-inertia of a rigid body when the conventional configuration (figure 1(a)) is impractical.Indeed, the dynamics of the swingpendulum and the conventional compound pendulum are identical, meaning that equation (3) applies to both cases (see figure 1, and section 2).In what follows, then, we test equation (3) empirically for the swing pendulum: first, by verifying that it correctly predicts T (sections 3 and 4); and second, by exploring how the equation can be used to infer I (section 5).Note that the experiments reported here were undertaken as 'furtherwork' exercises by undergraduate students during a first-year level physics laboratory on harmonic oscillations of rigid bodies.In this way our investigation of the swing-pendulum may be considered a case study in how students can be encouraged to engage enthusiastically with exploratory practical work, even in very simple contexts.

Theory
Before proceeding to the experiments, let us briefly confirm that the dynamics of the swingpendulum are identical to those of the conventional compound pendulum.To this end, consider the swing-pendulum depicted in figure 1(b), which is constructed by suspending a body of mass m from an external point O by light inextensible strings.If the distance from O to the body's centre-of-mass C is denoted h, and if θ is the angle between OC and the vertical, then the body will be subject to a restoring torque about O, and thence governed by Newton's second law for rotation [1] where I z is the body's moment-of-inertia about O. Thus, for small displacements satisfying the small angle approximation sin θ ≈ θ, the swingpendulum will oscillate harmonically according to (the frequency of oscillation) defines the oscillation time-period As claimed, therefore, the time-period of the swing-pendulum is identical to that of the conventional compound pendulum (see equation ( 1)), and can likewise be expressed using the parallel axis theorem (equation ( 2)) as where I is the moment-of-inertia of the body about its centre-of-mass C (see equation ( 3)).Indeed, since the light strings can be thought of as augmenting the body's boundary to incorporate O as an internal point, the conventional compound pendulum, and the compound swing-pendulum are physically equivalent systems.

Experiments
A simple way to confirm that equation (3) works for the swing-pendulum is to investigate how T varies with h for an object of known momentof-inertia I, and to this end we have chosen to base our experiments on metal bars (see figure 2).Indeed, a metal bar makes a suitable swingpendulum test-case for at least two reasons: first (i) it is not always easy to bore a hole through metal without specialist milling tools (so the conventional configuration with an internal pivot is not necessarily practical, or desirable); and second, (ii) a bar of length λ and thickness µ ≪ λ can be modelled as a thin uniform rod with moment-of-inertia and is therefore readily accessible to students from a theoretical perspective [1][2][3].
The basic apparatus for these experiments is shown in figure 2, where the swing-pendulum is constructed by suspending the bar from a screwhook O using household string, and h (the distance from O to the bar's centre-of-mass C) is adjusted by changing the string's length.Thus, for each value of h, it is possible to infer the swing-pendulum's time-period T by measuring the total time T N for some N oscillations, and computing T = T N /N [12].Note that that the oscillation amplitude must be small (say, less than 15 • [1]) to ensure that the small angle approximation sin θ ≈ θ holds (see section 2).
As detailed in table 1, our experiments considered two types of metal bar: a mild-steel boxsection bar; and an aluminium box-section bar.One advantage of box-section is that the string may be passed through the length of the bar internally, and fastened in a loop prior to suspension (as in figure 2); however, so long as the distance h can be measured accurately, any method of suspending the bar is acceptable (including asymmetric configurations, with the string secured externally), and students can be encouraged to explore different approaches [13].Note that the bars are considered 'thin' in the sense that both satisfy (µ/λ) 2 ≪ 1 (see table 1); similarly, the strings

Results
Data from our experiments on metal bars are listed in table 2, and plotted in figure 3. Assuming the bars to behave as thin, uniform rods, this data agrees well with the theoretical predictions of equation (3), i.e.
, where I = mλ 2 12 (10) may be computed using the values of λ and m in table 1 (see equation ( 9)).Hence, to within experimental uncertainty, we find as expected that the swing-pendulum behaves in the same way as a conventional compound pendulum.Note that it is a useful class exercise in dimensionless scaling [14] to have students express equation (3) in the normalised form where are the normalised time (T * ), distance (h * ), and moment-of-inertia (I * ) respectively.Indeed, by scaling the experimental data in this way (see table 2), both sets of measurements align with a single normalised theory curve (see figure 4).

Moment-of-inertia
In the preceding section we verified our expression for the time period T of equation ( 3) by using the fact that the moment-of-inertia was known to be I = mλ 2 /12 (see equation ( 9)).Observe, however, that if the moment-of-inertia of the body is not known, then equation ( 3) can be rearranged

to infer I from measurements of the time period T(h).
There are several methods for doing this [6], but perhaps the most straight-forward is simply to express equation (3) in the form so that I can be determined by computing its value from each measurement of the time-period T(h), and then taking an average.Indeed, here we  9) and ( 11)).
exploit this idea by using our scaled data to infer I from the swing-pendulum experiments by averaging over both sets of measurements simultaneously, that is, by using equation ( 13) in its dimensionless form

Conclusion
The compound 'swing-pendulum' is a novel approach to constructing a compound pendulum based on suspending a rigid body from an external pivot using light inextensible strings (section 1).By experimenting on metal rods, we have confirmed that the theory for such a swing-pendulum is consistent with that of a conventional (internally pivoted) compound pendulum (sections 2 and 3), and may similarly be used to infer the moment-of-inertia of a rigid body, especially in those situations when the conventional configuration is impractical (sections 4 and 5).In this way, the swing-pendulum offers a fresh perspective on the dynamics of the compound pendulum, and an alternative method for experimenting on moments-of-inertia in introductory physics laboratories.
When considered alongside more conventional studies of compound pendula, our swingpendulum experiments with first-year level undergraduate students have also presented several didactic benefits.For example, pivoting the system about an external point helps to emphasise the fact that the rotational characteristics of a body are not intrinsic properties of the body per se, rather that they are properties defined with respect to an appropriate reference axis.More importantly, however, the simplicity of the swingpendulum configuration makes it ideally suited for 'further-work' activities of the kind that encourage students to engage creatively in exploratory experimental work, and to think critically about experimental errors.Indeed, as we described in section 4, a particularly satisfying feature of our experiments on rods is the ease with which the data can be normalised for dimensionless scaling, a process which reveals important insights into symmetry and universality in physics [14].We look forward to developing these ideas to support work in other mechanical contexts, such as experiments on trifilar pendulums [16], as future investigations.
dynamics of coupled pendula suitable for distance teaching Phys.Educ.55 065008 [13] Most length measurements here were taken to an accuracy of ±0.005 m; however, the distance h was deduced from the string length using trigonometry, so subject to an uncertainty that increases as h decreases.Likewise, time measurements T N were taken to an accuracy of ±0.5 s, so that the uncertainty on T = T N /N varies depending on the total number of oscillations N considered

Figure 1 .
Figure 1.Two configurations for a compound pendulum: (a) the conventional configuration formed by pivoting a rigid body B about an internal point O; and (b) the novel 'swing-pendulum' configuration achieved by suspending the body B from an external pivot O using light strings slung around some points A and B.In both cases the centreof-mass C of the body is a distance h from O, where OC makes an angle θ with the vertical OP, and swings along an arc PC.Thus, the dynamics of each configuration are identical, with the weight of the body mg yielding a restoring force F = mg sin θ perpendicular to OC (see section 2).[Note: as indicated by the x-y-z reference axis, the z-axis passing through O is normal to the x-y plane containing PC].

Figure 2 .
Figure 2. Photograph (a) and schematic (b) of a swing-pendulum constructed from a steel bar of length λ and thickness µ suspended from a screw-hook O by household string, with h the distance from O to the bar's centre-ofmass C (see figure 1(b)).

[ 14 ]
Bissell J J, Ali A and Postle B J 2022 Illustrating dimensionless scaling with Hooke's law Phys.Educ.57 023008 [15] Barlow R J 1997 Statistics: a Guide to the Use of Statistical Methods in the Physical Sciences (Wiley) [16] Wang H 2021 A modified trifilar pendulum for simultaneously determining the moment of inertia and the mass of an irregular object Eur.J. Phys.42 015002

Table 1 .
Length λ, mass m, and thickness µ measurements for the metal bars.

Table 2 .
(11)sets of swing-pendulum data taken by students during introductory physics laboratories at the University of York[13].Here the fourth and fifth columns list the data normalised to λ and 2π √ λ/g respectively (see figures 3 and 4), with values for I * in the final column inferred according to equation(11)as I * = h * (T 2 * − h * ).Figure 3. Swing-pendulum time period T as a function of h for two metal bars: a steel bar (closed circles); and an aluminium bar (open circles).The theory curves are given by equation (10) assuming I = mλ 2 , with values for m and λ taken from table 1.