Page 1 of 19
European Journal of Applied Sciences – Vol. 12, No. 4
Publication Date: August 25, 2024
DOI:10.14738/aivp.124.17314.
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied
Sciences, Vol - 12(4). 164-182.
Services for Science and Education – United Kingdom
Harnessing Active Force: The Pumping Mechanism of Child’s
Swing Motion
Rong Li
Research Center for Industries of the Future, Westlake University, Hangzhou,
Zhejiang 310030, China and Key Laboratory of Coastal Environment and
Resources of Zhejiang Province, School of Engineering, Westlake University,
Hangzhou, Zhejiang 310030, China
Weicheng Cui
Research Center for Industries of the Future, Westlake University, Hangzhou,
Zhejiang 310030, China and Key Laboratory of Coastal Environment and
Resources of Zhejiang Province, School of Engineering, Westlake University,
Hangzhou, Zhejiang 310030, China
ABSTRACT
Life mechanics, an emerging field, focuses on the self-organizing motions
manipulated by the mind within living systems. This study introduces the concept
of 'active force’, generated by mind-body-environment interactions, as a
fundamental driver underlying these self-organizing movements. As an example,
we propose a new set of control equations to model the self-pumping swing
motion by incorporating the active force into Newton's second law. With this new
mechanical framework, we inversely derived the total (i.e., responsive) active
force due to the body-environment interaction from the child’s swing motions
with rapid standing and squatting movements. It revealed a pulse-like pattern of
the total active force along the swing length, driving changes in the radial speed
and swing length. This force counteracts the resistance and propels the swing,
which is not attainable by the stone. Consequently, the active force serves as the
foundational principle for self-organization in living systems, offering a novel
mechanical approach for understanding and predicting extraordinary movements
(e.g., sports and rehabilitation) regulated by the mind (e.g., nervous system) in
biological systems.
Keywords: Mind-body interaction, Active force, Newton's Second Law, Swing, Pulse
INTRODUCTION
Dyson, a physicist, remarked that the twenty-first century may be the century of biology [1].
Life, the most intricate of complex systems, is usually defined as a system that exhibits many
nontrivial movements, including responsiveness, energy transformation, metabolism, growth,
reproduction, and evolution [2]. Understanding these living movements poses the greatest
challenge in modern science [3, 4]. Accepting the axiom that force is the only reason for the
change in body movement, Schrödinger's seminal question "What is life?" [5] can be reframed
in the context of Newtonian mechanics: can we construct a mechanical model that describes
nontrivial movements inherent in life?
Page 2 of 19
165
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
Traditional Newtonian mechanics, which views the human body as the mechanical sum of its
parts, overlooks the complexity of mind-manipulating interactions and emergent behaviors.
From the late 19th century, it became clear that viewing life merely as a machine was
insufficient for understanding phenomena, such as embryonic cell development. For instance,
Driesch's experiments suggest that cells have an inherent ability to adapt to changing
environments [6, 7]. Further studies led to the birth of the modern system theory in the 1930s
[8]. The “vitality” and “entelechy” postulated by early vitalists [9] found modern
interpretation in the concept of “self-organization” in complex system science [10, 11].
However, despite the development of numerous phenomenological differential equations and
theories for complexities and lives [11-15], mechanical descriptions of the dynamics of self- determined (i.e., by mind) movement in life remain rare. This void in physicists’
understanding signals the need for the development of life mechanics [16].
In response to this need, we propose the concept of "active force”, an internal force and its
direct response arising from mind-body-environment interactions, as an integral part of life
mechanics. Recently, studying the autonomous motions in living and engineering systems has
led to a conceptual innovation related to “active” mechanics, for instance, “active matter” in
physics [17] and “active cell mechanics” in biomechanics [18], reminiscent the “action
potential” in electrophysiology [19]. Therefore, the active force is a straightforward
conceptual development along this concept series to define the mechanical deriver of
autonomous motions, which ranges from cells, fish, birds, and people. Indeed, the concept of
active force has been mentioned in a minority of literature, for instance, in experimental
analyses of certain skeletal and muscular motions [20, 21] as well as in active cell mechanics
[18], where it acts as the (stochastic) force driving self-organization behaviors. However, its
mechanical study is still in its infancy, possibly due to its vague mechanical definition,
calculation, and measuring complexity.
Therefore, there remains a substantial gap in understanding the generation mechanisms,
temporal patterns, and physiological significance of the active force. To fill this gap, our new
general system theory (NGST) [22-25] begins a preliminary study into its generation
mechanisms, classification, and mechanical representation in Newton’s laws. It presents
unified mechanics incorporating this active force from mind-body-environment interactions
and the passive forces arising from external interactions with other objects or the inanimate
matter-matter interactions independent of the mind. In other words, whether the force is
active or passive only depends on whether it is generated by mind-body interaction. This
classification doesn't violate Newton's laws; thus, the concept of active force is not a violation
but rather an extension of classical mechanics to incorporate the dynamics of living objects
under the living state. Our previous work on NGST has also shown that the active force must
compensate for the energy dissipated by resistance in a changing environment through doing
work [26, 27].
In this study, we examined the dynamics of a simple pendulum system to illustrate the
necessity of an active force to explain the observed phenomena in swing motions. Specifically,
by comparing the motion patterns of a child and a stone of equivalent weight, we demonstrate
that an active force that extends beyond the conventional forces of gravity and friction is
introduced by a child to generate motion patterns that diverge from those of a lifeless swing.
Page 3 of 19
Services for Science and Education – United Kingdom 166
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
This difference originates from the mind manipulating internal interactions and relative
motions between elements (e.g., leg muscles in swing motion or walk) of the body, which
induce a subsequent extra body-environment interaction, propelling the center of mass
motions of the swing. Our findings underscore the importance of incorporating an active force
into Newton's second law and the mechanical analysis of the mind-body-environment
interaction as a fundamental paradigm in active life mechanics.
GENERAL FRAMEWORK OF NEWTON'S SECOND LAW WITH ACTIVE FORCE
Consider the mechanics of a particle within a multiparticle system situated in an Earth-fixed,
non-inertial coordinate system (as depicted in Fig.1). This model is grounded in the
perception that our planet is in motion, an understanding that dates back to the era of Galileo
Galilei. In this context, the governing equation for each particle is derived from Newton's
second law:
d
2
dt
2 miri = Fi
P + Fi
A + Fi
D
, (1)
where mi and ri are the mass and displacement vectors of the ith particle, respectively, Fi
P
, Fi
A
and Fi
D
are the passive-driven force, active force, and dissipative force, respectively. Note that
the vectors are indicated by Roman letters.
Fig. 1: A schematic representation of a N-body system in an earth-fixed coordinate system. It
means that we only consider the motions of the particle systems relative to the observer on the
earth.
Eq. (1) embodies Newton's axiom that force is the agent of the motion change. The dynamics
of nonliving objects can be adequately described by the passive-driven forces Fi
P
and
dissipation forces Fi
D
. These passive-driven forces were generated from other objects,
including particles and the earth, in the system we studied. The most common passive-driven
forces in the macroscopic world are gravity (Fi
G = GmM/r
2
) and the static electromagnetic
force, that is, Fi
M = qi(E + vi × B), where E is the electronic field, B is the magnetic field
strength, qi and vi = ṙ
i are the charge and velocity of the particle, respectively. These forces
Page 4 of 19
167
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
can be expressed as a derivative of the generalized potential: Fi
P = −∇U + d(∂U/∂ṙ
i
)/dt,
where the generalized potential is U = VG + qφ − qA ∙ v. Here, VG is the gravity potential and
φ and A are the scalar and vector potentials of the electromagnetic field, respectively. This
analysis employs an Earth-fixed coordinate system and implicitly assumes the validity of
Newton's second law in a non-inertial coordinate system. According to the NGST ontology [26,
27], we must abandon the assumption of inertial coordinate systems because they do not exist
for human observers. However, the origin and expression of Fi
D
are typically complex. The
classical linear friction (Fi
D = −kivi
) and its corresponding Rayleigh dissipation function D =
∑ kivi
2
i /2 [28] is only a particular case (i.e., n = 1) of the general formula, that is, Fi
D = −kivi
n
.
Although complex, friction is always defined along the inverse direction of the velocity.
In addition, a new type of force, referred to as the "active force,” was introduced to explain the
initiating movement changes in living entities [22]. As introduced in the Introduction section,
the active force is the internal force directly generated by the mind-body interaction in a
living system. In contrast, the passive forces arise from external interactions with other
objects or the internal but inanimate matter-matter interactions independent of the mind.
Therefore, the critical distinction between active and passive forces is whether directly
generated by the mind-body interaction. Thus, any entity possessing a mind can exert an
active force; the separation of the mind from the body signifies the death of a living organism.
After death, a living object no longer exhibits an active force.
Thus, two immediate questions arise. How can the internal active force generate the center of
mass motions, and how can this active force be calculated? From our daily experiences,
humans exhibit active forces during their self-determined movements. For example, humans
and other animals can walk, run, and swing by using both the internal active force and the
body environment interactions. To generate these self-determined movements, one generally
utilizes the mechanism of mind-body-environment interaction: the mind issuing biosignal
(e.g., Nerve impulse) to modulate the interactions between elements of the body and their
relative motions. Subsequently, the body exerts extra force changes to the environment,
which in turn applies a reactive force that propels the body into the center of mass motion.
This tripartite interaction is essential to two kinds of active forces for the self-determined
movements of the living system. Compared to the baseline state of no motion or external
stimulation, we define the force change of the interactions between body elements as the
“internal active force” and the force change of the body-environment interaction as the
“responsive active force”. Thus, for each part of the body, we can obtain a decomposition of
the active force
Fi
A = Fi
I + Fi
R
, (2)
where Fi
I
is the internal active force and Fi
R
is the responsive active force. The mind-body
interaction mainly determines the former, while the environment constrains the latter.
Therefore, active force is interpreted as a mind-body-environment interaction. Without the
support of environment, the active force could not be generated.
Page 5 of 19
Services for Science and Education – United Kingdom 168
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
However, calculating these active forces is a significant challenge, as it is governed by the
unpredictable free will of the mind, as well as the complex mind-body-environment
interaction. We suggest initially adopting a model that explains this phenomenon qualitatively
and subsequently developing methods to quantify the active force. This study presented an
early effort and some initial results. In general, everyday human movements such as walking,
running, and swinging are inherently complex, necessitating rigorous modeling of the
interactions and movements of various body parts [29-31]. However, this paper aims not to
explore modeling too many complex motions but to clarify the differences between active
forces in living systems and those in inanimate bodies. Therefore, our primary focus in this
study was on the motion of the center of mass for its simplicity of degree of freedom and the
more straightforward measurement of the responsive active force compared to the internal
active force. In this case, the internal active forces of different body parts cancel with each
other (i.e., ∑ Fi
I
i = 0 following Newton’s third law), while the summation of the responsive
active force from the environment to the body is nonzero. Thus, in line with the definition of
the center of mass coordinates (R = ∑i miri / ∑i mi
), Newton’s second law can be obtained
from the summation of Eq. (1) as,
M
d
2
dt
2
R = Fp + Fa + f, (3)
where M = ∑i mi
is the total mass, Fp = ∑ Fi
P
i
is total passive (environment-to-body) force at
the baseline state of the body, f = ∑ Fi
D
i
is the total passive-driven force, total active-driven
force, and the total dissipation force, respectively. Fa = ∑ Fi
A
i = ∑ Fi
R
i
is the total active force
of the body relative to the baseline state, equaling the total responsive active force since
∑ Fi
I
i = 0. In this paper, we employ Eq. (2) to extract the total (responsive) active force
driving the swing's motion, deferring the elucidation of internal active forces to future
investigations.
THE RESPONSIVE ACTIVE FORCE FOR PUMPING A SWING BY A LIVING SYSTEM
This section explores how an active force propels a swing, mainly by comparing the dynamic
differences (displacement, velocity, and force) between the swing motions of a child and a
stone. This comparison enables us to quantitatively derive the temporal pattern of the
responsive active force, revealing the essential role of active force in explaining the motions of
living systems.
The Governing Equation for Pumping A Swing
Applying Eq. (3) elucidates the differences between the swing motions of a child and a stone,
considering only simple pendulum motions within a vertical two-dimensional plane. Fig. 2
illustrates the simple pendulum system: (a) represents a classical case with a non-living stone,
whereas (b) substitutes the stone with a child of equivalent weight. In the stone case,
L denotes the length of the rigid massless rod, m denotes the mass of the stone, and θ
represents the angle of the rod along the vertical axis. Furthermore, three forces acted at the
center of the mass of the stone. First, gravity, Fg = mg acts in the downward vertical direction.
Second, a passive tension force, Tp acts along the rod owing to the balance of gravity and
centrifugal force toward the frictionless pivot. Third, a friction force f resists the motion of
Page 9 of 19
Services for Science and Education – United Kingdom 172
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
37]. It can be considered as a coupled oscillator system composed of a swing and a human.
Typically, there are two pumping strategies: pumping from a standing position (as depicted in
Fig. 3) and a seated position. In the former instance, the person stands at the lowest point and
crouches at the highest point during the swing motion. Each stand-crouch cycle enhances the
swing amplitude. The analysis demonstrated that each crouch-stand cycle provides a swing
with an energy boost from the rider. In the latter scenario, the person abruptly rotates their
body around the end of the swing chain. The amplitude of the swing increases as these
rotations elevate the rider slightly above the highest level.
FIG. 3: Strategy for pumping a swing while standing, adapted from Ref. [38]. The child stands up
near the lowest point and crouches down near the highest point during the swing motion.
In the child’s swing motion, the mind determines the pumping strategies, while the active
force derives the body to motion, and environmental constraints provide the control
conditions for this driver; all these three factors of mind, active force and environmental
constraints are paramount. However, prior research on swing mechanics has primarily
concentrated on effective pumping strategies and environmental constraints, such as the
pumping mode (standing or seated [34, 38]), the modulation of frequency, and the initial
phase [37] under the swing constraints. On the other hand, the force underlying swing
pumping, particularly the active force that drives these movements, has been less explored.
This indicates that the essential force mechanism generated by the mind-body-environment
interaction— has not been adequately addressed. Our study aims to fill this gap by focusing
on the (internal or responsive) active force, which directly results from the mind-body- environment interaction and serves as the driver of the swing's active motion. This work
mainly focuses on the responsive active force, representing the body-environment
interaction. Understanding the temporal evolution pattern of this responsive active force is
crucial, as it not only reveals the direct capacity of individuals to drive the center of mass
movements but also how the environment responds to the body. Given that the force pattern
corresponding to the stand-crouch motion is simpler, we mainly concentrate on pumping
from the standing position in this study.
Page 11 of 19
Services for Science and Education – United Kingdom 174
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
δ(θ̇) =
d
dθ̇
H(θ̇) =
1
cb
1
2 + e
−
θ̇
cb + e
θ̇
cb
, (16 − d)
Here, ca determines the temporal pattern of standing up at the minimum angle, whereas cb
determines the temporal pattern of crouching down at the maximum angle. These two
parameters can be independently chosen according to the child’s free will.
The Responsive Active Force for Pumping A Swing Without Friction
By employing Eqns. (12)–(16), we can derive the time evolution of several parameters: the
angle θ(t), responsive active force Fa
∥
, pendulum length L(t) = L0 + l(t), vertical height of the
mass center of the child h(t) = L0 − L(t) cos θ, the circumferential velocity v⊥ = L(t)θ̇(t) and
radial velocity v∥ = l(̇ t). The angle and velocities in two directions depict the dynamics of the
motion, whereas the alterations in swing length and height represent the spatial state. The
responsive active force, which is our primary objective, emerged from these calculations.
To execute this simulation, we must establish the parameters within these equations and
initial conditions. Let us assume that the masses of the individual and stone are identical, that
is, m = 20 kg. The swing's maximum pendulum length, L0, was 2 m, and the gravitational
acceleration g was 9.81 m/s
2
. We assume the height change of the child is k = 0.2m and zero
for the stone. The friction coefficient b, is influenced by the friction between the individual
and air, the swing, and between the swing rope and the fixed point.
Fig. 4: Active-motion simulations using Eqns. (12-16) for a child pumping a swing without
friction, as depicted by the solid red lines. In contrast, the dashed black lines represent the
simple pendulum motion of a stone with the same mass as the child. The parameters selected
for these simulations are outlined in Table I.
In our study, we account for the linear frictional force at low speeds between the child (or
stone) and the air, represented as f = −bv. The coefficient b is acknowledged to be variable,
Page 12 of 19
175
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
contingent upon the changing conditions of interaction between the child (or stone) and the
air. Consequently, b is treated as a free adjustable parameter, facilitating a simplified and
flexible model that can adapt to a spectrum of realistic scenarios. The coefficients ca and cb,
which dictate the patterns of standing and crouching, as well as the initial angle θ(0) and
initial velocity θ̇(0) can be freely chosen by the child’s free will. Consequently, in this study,
we treat parameters b, ca, cb, θ(0) and θ̇(0)) as free adjustable parameters.
By substituting these parameters into Eqns. (12)– (16), we can juxtapose the swings of the
stone and the child. The simulations for the frictionless scenario are shown in Figure 4. As is
commonly understood, the stone (represented by dashed black lines) performs harmonic
oscillation, maintaining a constant amplitude throughout each period. Conversely, the child
(denoted by solid red lines) generates an enhanced oscillation, with the amplitude increasing
in each period. The figures show that the augmentation in the amplitudes of the angle (a) and
circumferential velocity (b) aligns with the continuous increase in radial velocity (e),
prompted by the child's periodically increasing responsive active force (d). This responsive
active force emerged as four pulses per cycle, corresponding to two standing and two
crouching instances. More specifically, during the first 1/4 period, the initial angles (a) and
initial circumferential velocities (b) for the stone and child were close. However, the child’s
swift standing motion around (see (d)-(f)) leads to a gradual increase in the angle and the
circumferential velocity, reaching peak differences from the stone around 2.0 s and 2.2 s.
Consequently, we conclude that in swing motion, the child's active force stimulates the
amplification of the swing's amplitude.
Table I: Parameters selection for the stone (zero active force) and the child (finite
active force) with zero friction.
System Mass Maximum
length
Gravitational
acceleration
Friction
coefficient
Standing
height
Pulse
angle
Pulse
velocity
Initial
angle
Initial
velocity
M(kg) L0(m) g(kg. s
−2
) b(kg. s
−1
) k ca cb(s
−1
) θ(0) θ̇(0)
Stone 20 2 9.8 0 0 — — π/6 0
Child 20 2 9.8 0 0.2 0.12 0.20 π/6 0
We proceeded with a quantitative analysis of the energy-pumping mechanism facilitated by
the responsive active force. In a frictionless environment, we assume that standing up and
crouching down are executed instantaneously. Because the responsive active force solely
impacts the radial motion, neither the circumferential momentum (or velocity) nor the angle
changes this process. The critical variables that vary are the pendulum length, which shifts
from L0 to L0 − k, and the radial velocity v∥ = l(̇ t). By examining the nth instance of standing
up, we can establish a relationship between the maximum swing amplitudes before (θn−1)
and after (θn) the actions. The law of energy conservation dictates that the circumferential
kinetic energy (m(v⊥
n
)
2
/2) at the lowest point originates from the potential energy decrease
(mgL0
(1 − cosθn−1
)) of the child from the maximum swing angle θn−1. After standing up, this
kinetic energy transitions into a potential energy increase (mg(L0 − k)(1 − cosθn
)) of the
maximum swing θn. As such, the potential energy decreases before standing up is equal to the
increase following it,
Page 13 of 19
Services for Science and Education – United Kingdom 176
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
mgL0
(1 − cosθn−1
) ≈ mg(L0 − k)(1 − cosθn
). (17)
From this relation, we can obtain the nth maximum angle, height, and velocity as:
θn ≈ arccos [
cosθn−1 − k
′
1 − k
′
], (18)
where k
′ = k/L0. Eq. (18) reveals that the swing amplitude increases every half-cycle
according to a function related to 1/(1 − k
′
). This reveals that the swing accumulates the net
energy from the child’s active motions in each pumping cycle. This net energy stems from the
work done by the child’s active force when standing up, which increases the height by k, over
the energy spent when squatting down, which decreases the height by kcosθn. It is worth
mentioning that Eq. (18) is obtained based on the neglect of the impact of standing squatting
motions. This contribution is −2l(̇ t)θ̇(t)/L(t) for θ̈(t) in Eq. (13), revealing that the standing
results in an increase for |θ̇(t)|. Therefore, the realistic increase of θn (shown in Fig. 4) should
be greater the prediction θn = arccos[1.11(cosθn−1 − 0.1)] by substituting parameters in the
numerical simulations into Eq. (18).
The Responsive Active Force for Pumping A Swing with Linear Friction
Next, we explore a more realistic scenario in which both active and frictional forces are at
play. A particularly noteworthy situation occurs when the responsive active force
counterbalances the frictional force, leading to stable oscillation with a constant amplitude
that neither decays nor expands. Such a state represents a stable equilibrium that is
sometimes observed in swing sports. An intriguing question is whether our active-force
model (Eq. (12) and (13)) can accurately simulate this condition.
Fig. 5: Active-motion simulations using Eqns. (12-16) showcasing a child pumping a swing in
the presence of finite friction, as shown by the solid red lines. To serve as comparisons, the
solid blue lines (indicating finite friction) and dashed black lines (representing no friction)
Page 14 of 19
177
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
illustrate the simple pendulum motion of a stone with the same mass as the child. The
parameters used for these simulations are listed in Table II.
As shown in Fig. 5, the active-force model successfully replicates this scenario. Specifically, for
the stone without the active force, the swing amplitude demonstrated a decay (as shown by
the solid blue lines) when compared to the harmonic oscillations without friction
(represented by dashed black lines). In contrast, for a child exerting an appropriately strong
responsive active force (denoted by solid red lines), the swing amplitude remains constant (as
seen in images (a) to (c)). This essentially means that, in this situation, the energy pumped
from the responsive active force successfully counteracted the damping effect of friction.
Initially, it can be inferred that the frictional force is counterbalanced by the responsive active
force, implying that only the net active force minus the frictional force needs to be considered.
However, this was not the case. As is evident from Fig. 5, notable differences exist between the
balance of the active force and frictional contributions to the swing amplitude (solid red lines)
and the simple harmonic motion, where both forces perfectly nullify each other (dashed black
lines). These discrepancies are manifested in the smaller period of the former case (2.82 s) by
2% compared with the latter case (2.89 s), despite a less than 1/500 difference between their
amplitudes. This is intriguing, as a larger amplitude generally corresponds to a larger period
in a scenario devoid of an active force. Nevertheless, according to the simple relation T =
2π√L/g, the nontrivial smaller period for the child is brought about by the reduction in the
pendulum length owing to the standing squatting motions. The root cause of this is the
frictional force comprising both the radial and circumferential components, in contrast to the
active force, which is exclusively radial. This results in the impracticality of perfectly
counteracting the frictional force regardless of the form of the radial active force. The actual
scenario involves achieving an energy balance with the radial work executed by the active
force compensating for the energy loss induced by the circumferential component of the
frictional force.
Table II: The parameters selected for the swing simulations involving a stone (zero
active force) and a child (finite active force) under conditions of finite friction, as
shown in Fig. 5.
System Mass Maximum
length
Gravitational
acceleration
Friction
coefficient
Standing
height
Pulse
angle
Pulse
velocity
Initial
angle
Initial
velocity
M
(kg)
L0(m) g(kg. s
−2
) b(kg. s
−1
) k ca cb(s
−1
) θ(0) θ̇(0)
Stone 20 2 9.8 3.64 0 — — π/6 0
Child 20 2 9.8 3.64 0.2 0.12 0.20 π/6 0
We now turn to the primary parameters that determine the amplitude of the swing propelled
by the active force. Holding constant parameters such as the child's mass, swing length,
resistance, and initial conditions, we found that the child's center of mass elevation (the
height difference between the standing and squatting positions) significantly influences the
amplitude. For instance, Fig. 6(a) illustrates that when the standing amplitude surpasses 0.2
meters, the oscillation of the swing gradually increases (as indicated by the blue lines).
Conversely, the oscillation diminishes progressively when the amplitude drops below 0.2
Page 15 of 19
Services for Science and Education – United Kingdom 178
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
meters (indicated by dashed black lines). On the other hand, when the difference in the center
of mass height between standing and squatting is fixed at 0.2 meters for each swing (Fig. 6f),
the speed of standing (ca
) and squatting (cb) also affects the amplitude (Fig. 6d). The faster
the standing and squatting motions are, the larger the magnitudes are. Therefore, beyond the
typical factors influencing both animate and inanimate systems, such as mass, swing length,
resistance, and initial conditions, the pivotal determinant of the amplitude in a swing
propelled by standing is the difference in the center of mass height between standing and
squatting and the standing and squatting speeds.
Fig. 6: The amplitude alterations attributable to height (k) variation ((a)-(c)) or speed (ca and
cb) changes in the standing-squatting motion ((d)-(f)). The adjusted parameters are denoted in
legends. Other parameters align with those specified for active motions in Table II.
DISCUSSION AND IMPLICATION
This paper introduces a compelling argument for the existence and necessity of the 'active
force' concept to elucidate the dynamics of living objects in the living state, using the human- pumping swing action as an example. This active force, originating from the interactions
among the body, mind, and environment, provides a unique contribution to the dynamics of
living systems, a facet that conventional physics often disregards. We established a
comprehensive mechanical framework applicable to living and nonliving systems by
incorporating an active force into Newton's second law. This innovative governing equation
was applied to describe the swing motions of a child and an equally weighted stone, focusing
on their differences. Our study reveals that children with active force can sustain or augment
their amplitude despite friction, whereas a stone's oscillation can only decrease. Quantitative
analysis indicated that the energy pumping caused by the responsive active force (i.e., the
total active force of the body) performing work in each standing motion was responsible for
this phenomenon.
Furthermore, we identified that the responsive active force exhibits a pulse-like pattern,
whereas a linear style (e.g., l = −kθ) fails to generate a self-excited oscillation. Therefore,
Page 16 of 19
179
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
beyond the typical factors influencing both animate and inanimate systems, such as mass,
swing length, resistance, and initial conditions, the pivotal determinant of the amplitude in a
swing propelled by standing is the difference in the center of mass height between standing
and squatting and the standing and squatting speeds. These findings underscore the existence
and necessity of the 'active force' concept for understanding living system dynamics, offering
a paradigm and mechanical framework for further exploration into life mechanics [16].
However, this research constitutes a preliminary study of the global (responsive) active force.
Future research should first probe further into the quantitative measurements and
calculations of both internal (i.e., local) and responsive active forces in the self-determined
motions of living beings, such as development, metabolism, and sports in plants, animals, and
humans. Such efforts may reveal novel patterns and mechanisms of both active forces as well
as their critical quantitative relations, laying a scientific foundation for predicting and
controlling life movements. Furthermore, although the current model prioritizes the active
force to emphasize its significance, it does not neglect the effects of swing constraints but can
address the impacts of both body and environmental constraints. More importantly, by
studying the quantitative mechanical relation between the internal and the responsive active
forces in future research, we can clarify the physical mechanism of the environmental
constraints or, in other words, the body-environment interaction.
It is crucial to remember that the active force discussed herein represents the summation of
all the active forces experienced by the human body's center of mass. Actual measurements
might necessitate measuring the contact at multiple points between the human body and the
interacting object, such as the contact of the feet and hands with a swing. By vectorially
synthesizing the forces at multiple points, we can obtain the resultant force at the center of
mass, as discussed in this paper. Consequently, further theoretical development should
consider the movement of multiple body parts under forces acting at various points on the
human body via Eq. (1). In this context, an immediate question would be to investigate the
active force pattern in 'pumping a swing from a seated position’ and compare it with the
results of the present study. This exploration would necessitate considering three subsystems
of the human body: the torso, thighs, and lower legs [25,31].
Particularly noteworthy is that the active force is a pivotal control mechanism of the mind
over the body during human motion. Optimal movement control is often characterized by the
timeliness, strength, and controllability of the active force in response to the mind. Therefore,
the practice and measurement of the active force may be a significant topic in the field of
sports training and rehabilitation therapy. In this regard, the relationships between active
force, bioelectricity, and consciousness present an intriguing topic. The active forces shown in
Figs. (4)–(6) exhibit a pulsating characteristic in the time series, reminiscent of the pulsation
patterns of bioelectric signals such as ECG and EEG [39]. We hypothesized that this similarity
is not coincidental, suggesting that active force indeed embodies the characteristics of life.
This is because the active force pattern in the pump swing is a consequence of an individual's
conscious decision in their movement mode, which is a product of the brain and heart's
electrical activities. In future studies, we will propose simultaneous measurements and
mathematical modeling of active force with EEG and ECG signals to investigate their
Page 17 of 19
Services for Science and Education – United Kingdom 180
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
quantitative correlations, potentially advancing our understanding of the underlying physical
mechanisms connecting active force, bioelectricity, and human conscious movement.
ACKNOWLEDGMENT
This research was funded by the National Key Research and Development Program of China
(2022YFC2805200), start-up funding from Westlake University (041030150118), and the
Scientific Research Funding Project of Westlake University (2021WUFP017). During the
preparation of this work the authors used the GPT-4 model in order to do language editing.
After using this tool, the authors reviewed and edited the content as needed and take full
responsibility for the content of the publication.
Data Access Statement
All relevant data are within the paper.
Conflict of Interest Declaration
The authors have declared that no competing interests exist.
References
1. Dyson, F., Our biotech future. The New York Review of Books, 2007. 54: p. 12.
2. Lynn, M., Carl, S., & Dorion, S. (2023). Life. Retrieved April 6, 2023, from
https://www.britannica.com/science/life.
3. Kennedy, D. and C. Norman, What Don't We Know? Science, 2005. 309(5731): p. 75-75.
4. Bianconi, G., et al., Complex systems in the spotlight: next steps after the 2021 Nobel Prize in Physics. J. Phys.
Complexity, 2023. 4(1): p. 010201.
5. Schrödinger, E., What is life?: With mind and matter and autobiographical sketches2012: Cambridge
University Press.
6. Allen, G., Life sciences in the twentieth century. J. Hist. Biol., 1976. 9(2).
7. Allen, G.E., Mechanism, vitalism and organicism in late nineteenth and twentieth-century biology: the
importance of historical context. Stud. Hist. Phi. Part C, 2005. 36(2): p. 261-283.
8. Von Bertalanffy, L., The history and status of general systems theory. Acad. Manage. J., 1972. 15(4): p. 407-
426.
9. Driesch, H., Psychical research: The science of the super-normal1933: Ayer Publishing.
10. Von Foerster, H. and G.W. Zopf. Principles of self-organization. in University of Illinois Symposium on Self- Organization,(1961: Robert Allerton Park). 1962. Pergamon Press.
11. Haken, H., Synergetics: Introduction and Advanced topics2004, Berlin: Springer
12. Bertalanffy, L.v., General system theory: Foundations, development, applications1968: G. Braziller.
13. Royama, T., Analytical population dynamics1992, London: Chapman & Hall.
14. Castellani, B and Gerrits, L . Map of the complexity sciences. Art and Science Factory, LLC. (2021).
Page 18 of 19
181
Li, R., & Cui, W. (2024). Harnessing Active Force: The Pumping Mechanism of Child’s Swing Motion. European Journal of Applied Sciences, Vol -
12(4). 164-182.
URL: http://dx.doi.org/10.14738/aivp.124.17314
15. Kelso, J.A.S., On the coordination dynamics of (animate) moving bodies. Journal of Physics: Complexity, 2022.
3(3): p. 031001.
16. Yang, W., P. Zhao, and W. H.T., An overview of mechanics (in Chinese)2020, Beijing: Science Press
17. Gompper, G., et al., The 2020 motile active matter roadmap. J. Phys.: Condens. Matter 2020. 32(19): p.
193001.
18. Ahmed, W.W., É. Fodor, and T. Betz, Active cell mechanics: Measurement and theory. BBA - Mol. Cell Res.,
2015. 1853(11, Part B): p. 3083-3094.
19. HILL, A.V., The Mechanism of the Nervous System. Nature, 1928. 123(3088): p. 9-11.
20. Rassier, D.E., B. MacIntosh, and W. Herzog, Length dependence of active force production in skeletal muscle. J.
Appl. Physiol. , 1999. 86(5): p. 1445-1457.
21. Weiss, P.L., R.E. Kearney, and I.W. Hunter, Position dependence of ankle joint dynamics—II. Active mechanics.
J. Biomechanics, 1986. 19(9): p. 737-751.
22. W. C. Cui, On an axiomatic foundation for a theory of everything. Philos. Study, 2021. 11(4): p. 241-267.
23. Weicheng, C., On the philosophical ontology for a general system theory. Philos. Study, 2021. 11(6): p. 443-
458.
24. Cui, W.C., R. Li, and L.L. Pan, A comparison of new general system theory philosophy with Einstein and Bohr.
Philos. Study, 2023. 13: p. 1-22.
25. Weicheng, C., On the trajectory prediction of a throwing object using new general system theory. Philos. Study,
2022. 12(2): p. 53-64.
26. Pan, L. and W. Cui, Re-examination of the two-body problem using our new general system theory. Philos.
Study, 2021. 11(12): p. 891-913.
27. Yuke, H. and C. Weicheng, Effect of the Earth Movement on the Measured Moon Trajectory. Philos. Study,
2022. 12: p. 641-680.
28. Goldstein, H., C. Poole, and J. Safko, Classical mechanics: 3rd2002: Pearson Education
29. Klimina, L. and A. Formalskii, Three-link mechanism as a model of a person on a swing. J. Comput. Syst. Sci.
Int., 2020. 59: p. 728-744.
30. Onyshko, S. and D. Winter, A mathematical model for the dynamics of human locomotion. J. Biomech., 1980.
13(4): p. 361-368.
31. Hanavan, E.P., A mathematical model of the human body. Dayton, OH: Aerospace Medical Division, Wright
Patterson Air Force Base, 1964: p. 64-102.
32. Gore, B.F., The Child's Swing. Am. J. Phys., 1970. 38(3): p. 378-379.
33. Curry, S.M., How children swing. Am. J. Phys., 1976. 44(10): p. 924-926.
34. Case, W.B., The pumping of a swing from the standing position. Am. J. Phys., 1996. 64(3): p. 215-220.
35. Bae, S., Equivalence of the pumping of a swing and the parametric resonance. Eur. J. Phys., 2006. 27(2): p.
Page 19 of 19
Services for Science and Education – United Kingdom 182
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 4, August-2024
291.
36. Glendinning, P., Adaptive resonance and pumping a swing. Eur. J. Phys., 2020. 41(2): p. 025006.
37. Hirata, C., et al., Initial phase and frequency modulations of pumping a playground swing. Phys. Rev. E, 2023.
107(4): p. 044203.
38. Wirkus, S., R. Rand, and A. Ruina, How to pump a swing. Coll. Math. J., 1998. 29(4): p. 266-275.
39. Malmivuo, J. and R. Plonsey, Bioelectromagnetism: principles and applications of bioelectric and biomagnetic
fields1995: Oxford University Press.